Drorbn - User contributions [en]

1617-257/TUT-R-12

2016-12-01T23:32:26Z

Jeffim:

Comments from the tutorial on 12/1/16.

Someone asked about how to do a problem from the currently assigned problem. A suggestion for how to show the integral exists was by using Fubini's theorem. I OK'd this idea in the tutorial, but it is actually not OK. In order to use Fubini's theorem, one needs to know that the function in question is integrable (and that's exactly what the problem asks to do). I'm going to go with my original suggestion and suggest making estimates using some compact and rectifiable exhaustion for this function to do this question.

Another person gave an argument for how to do 15-8(b) which works fine if the function is continuous everywhere, but since there function may not necessarily be continuous everywhere, you do in fact need to use that it's locally bounded.

Sorry about this missteps.

1617-257/TUT-R-12

2016-12-01T23:32:05Z

Jeffim:

Comments from the tutorial on 12/1/16.

Someone asked about how to do a problem from the currently assigned problem. A suggestion for how to show the integral exists was via Fubini's theorem. I OK'd this idea in the tutorial, but it is actually not OK. In order to use Fubini's theorem, one needs to know that the function in question is integrable (and that's exactly what the problem asks to do). I'm going to go with my original suggestion and suggest making estimates using some compact and rectifiable exhaustion for this function to do this question.

Another person gave an argument for how to do 15-8(b) which works fine if the function is continuous everywhere, but since there function may not necessarily be continuous everywhere, you do in fact need to use that it's locally bounded.

Sorry about this missteps.

1617-257/TUT-R-12

2016-12-01T23:29:44Z

Jeffim: Created page with "Comments from the tutorial on 12/1/16. Someone asked about how to do a problem from the currently assigned problem. A suggestion for how to show the integral exists was via F..."

1617-257

2016-12-01T23:25:42Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257a-AnalysisII/|1617-257a notebook}}.

* Some [http://drorbn.net/bbs/show?prefix=1617-257 blackboard shots].

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

* [[1617-257/TUT-R-8| Tutorial R-8 comments]]

* [[1617-257/TUT-R-12| Tutorial R-12 comments]]

* [[1617-257/HW_formatting| HW formatting comments]]

1617-257

2016-12-01T23:25:30Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257a-AnalysisII/|1617-257a notebook}}.

* Some [http://drorbn.net/bbs/show?prefix=1617-257 blackboard shots].

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

* [[1617-257/TUT-R-8| Tutorial R-8 comments]]

* [[1617-257/TUT-R-8| Tutorial R-12 comments]]

* [[1617-257/HW_formatting| HW formatting comments]]

1617-257/HW formatting

2016-11-30T22:09:48Z

Jeffim:

'''Comments about labeling'''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. </li>

<li> Please take the time to legibly write your full name on the top of the paper in the following order: [first name] [last name]. </li>

</ul>

'''Comments about writing'''

<ul>

<li> Take the time to tidy up your arguments and write them out neatly (if you can't write neatly like me, learn to type your assignments). </li>

<li> If your homework assignments are difficult to staple, you are probably showing too much unnecessary work or doing work which isn't necessary. </li>

</ul>

'''General comments/requests'''

<ul>

<li> Arguing with me about every little point on homework assignments is not going to make or break your grade in this class. I really do not think this is a good use of time. </li>

<li> Please take the time (up to 9 days after the assignment was due) to carefully think about why you might have lost points on an assignment/test before consulting me, Jihad, or the professor. I don't want to discourage people from discussing things with me, but it's been somewhat frustrating to see many people come straight to me without considering the possibility that they may wrong. Out of all of the reviews (well over 50) I've made this year, I can count on one hand the number of times there was a legitimate error on the grader's end.
</li>

</ul>

Best wishes with your exams. Hope you all have a relaxing break. 
Jeff

1617-257/HW formatting

2016-11-30T22:07:37Z

Jeffim:

'''Comments about labeling'''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. </li>

<li> Please take the time to legibly write your full name on the top of the paper in the following order: [first name] [last name]. </li>

</ul>

'''Comments about writing'''

<ul>

<li> Take the time to tidy up your arguments and write them out neatly (if you can't write neatly like me, learn to type your assignments). </li>

<li> If your homework assignments are difficult to staple, you are probably showing too much unnecessary work or doing work which isn't necessary. </li>

</ul>

'''General comments/requests'''

<ul>

<li> Arguing with me about every little point on homework assignments is not going to make or break your grade in this class. I really do not think this is a good use of time. </li>

<li> Please take the time to carefully think about why you might have lost points on an assignment/test before consulting me, Jihad, or the professor. I don't want to discourage people from discussing things with me, but it's been somewhat frustrating to see many people come straight to me without considering the possibility that they may wrong. Out of all of the reviews (well over 50) I've made this year, I can count on one hand the number of times there was a legitimate error on the grader's end.
</li>

</ul>

Best wishes with your exams. Hope you all have a relaxing break. 
Jeff

1617-257/HW formatting

2016-11-30T22:01:13Z

Jeffim:

'''Comments about labeling'''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. </li>

<li> Please take the time to legibly write your full name on the top of the paper in the following order: [first name] [last name]. </li>

</ul>

'''Comments about writing'''

<ul>

<li> Take the time to tidy up your arguments and write them out neatly (if you can't write neatly like me, learn to type your assignments). </li>

<li> If your homework assignments are difficult to staple, you are probably showing too much unnecessary work or doing work which isn't necessary. </li>

</ul>

'''General comments/requests'''

<ul>

<li> Arguing with me about every little point on homework assignments is not going to make or break your grade in this class. I really do not think this is a good use of time. </li>

<li> Please take the time to carefully think about why you might have lost points on an assignment/test before consulting me, Jihad, or the professor. I don't want to discourage people from discussing things with me, but it's been somewhat frustrating to see many people come straight to me without considering the possibility that they may be wrong.
</li>

</ul>

Best wishes with your exams. Hope you all have a relaxing break. 
Jeff

1617-257/HW formatting

2016-11-30T21:50:17Z

Jeffim:

'''Comments about labeling'''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. </li>

<li> Please take the time to legibly write your full name on the top of the paper in the following order: [first name] [last name]. </li>

</ul>

'''Comments about writing'''

<ul>

<li> Take the time to tidy up your arguments and write them out neatly (if you can't write neatly like me, learn to type your assignments). </li>

<li> If your homework assignments are difficult to staple, you are probably showing too much unnecessary work or doing work which isn't necessary. </li>

</ul>

'''General comments/requests'''

<ul>

<li> Arguing with me about every little point on homework assignments is not going to make or break your grade in this class. I really do not think this is a good use of time. </li>

<li> Please take the time to carefully think about why you might have lost points on an assignment/test before consulting me, Jihad, or the professor. I don't want to discourage people from discussing things with me, but it's been somewhat frustrating to see that many people have not thought at all about what might have gone wrong and simply come straight to me to explain exactly what went wrong.
</li>

</ul>

Best wishes with your exams. Have a good break! 
Jeff

1617-257/HW formatting

2016-11-30T21:49:46Z

Jeffim:

'''Comments about labeling'''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. </li>

<li> Please take the time to legibly write your full name on the top of the paper in the following order: [first name] [last name]. </li>

</ul>

'''Comments about writing'''

<ul>

<li> Take the time to tidy up your arguments and write them out neatly (if you can't write neatly like me, learn to type your assignments). </li>

<li> If your homework assignments are difficult to staple, you are probably showing too much unnecessary work or doing work which isn't necessary. </li>

</ul>

'''General comments/requests'''

<ul>

<li> Arguing with me about every little point on homework assignments is not going to make or break your grade in this class. I really do not think this is a good use of time. </li>

<li> Please take the time to carefully think about why you might have lost points on an assignment/test before consulting me, Jihad, or the professor. I don't want to discourage people from discussing things with me, but it's been somewhat frustrating to see that many people have not thought at all about what might have gone wrong and simply come straight to me to explain exactly what went wrong.
</li>

</ul>

Best wishes with your exams and happy holidays.

Jeff

1617-257/HW formatting

2016-11-30T21:47:58Z

Jeffim:

'''Comments about labeling'''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. </li>

<li> Please take the time to legibly write your full name on the top of the paper in the following order: [first name] [last name]. </li>

</ul>

'''Comments about writing'''

<ul>

<li> Take the time to tidy up your arguments and write them out neatly (if you can't write neatly like me, learn to type your assignments). </li>

<li> If your homework assignments are difficult to staple, you should be asking yourself if you are showing too much unnecessary work or doing work which isn't necessary. </li>

</ul>

'''General comments/requests'''

<ul>

<li> Arguing with me about every little point on homework assignments is not going to make or break your grade in this class. I really do not think this is a good use of time. </li>

<li> Please take the time to carefully think about why you might have lost points on an assignment/test before consulting me, Jihad, or the professor. I don't want to discourage people from discussing things with me, but it's been somewhat frustrating to see that many people have not thought at all about what might have gone wrong and simply come straight to me to explain exactly what went wrong.
</li>

</ul>

Best wishes with your exams and happy holidays.
Jeff

1617-257/HW formatting

2016-11-30T21:35:35Z

Jeffim:

1617-257/HW formatting

2016-11-30T21:28:08Z

Jeffim: Created page with "''Comments about labeling'' <ul> <li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li> <li> If (and only if) yo..."

''Comments about labeling''

<ul>

<li> If you do not pick up your HW at tutorials, please do not label your HW with one of the tutorial times. </li>

<li> If (and only if) you want to pick up your HW at tutorials, please use only the following labels: the TA's name/initials, Wednesday, or Thursday. Please do not write some weird number which I am unable to decrypt. </li>

</ul>

''Comments about writing''

1617-257

2016-11-30T21:22:10Z

Jeffim:

1617-257/TUT-R-8

2016-11-07T20:36:27Z

Jeffim:

On 11/3/16, we discussed some questions from the exam:

'''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</math> has a limit point.

''Proof''. If <math>E</math> has no limit points, then <math>E</math> is a closed subset of a compact space and is therefore compact in itself. Since each point of <math>E</math> is isolated, we may find for each point <math>e \in E</math> a neighborhood <math>U_e</math> such that <math> E \cap U_e = \{ e \}</math>. The collection <math>\{ U_e\}_{e \in E}</math> is an open cover of E which clearly has no finite subcover.

'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.

''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B_1(0)</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is closer to <math>z</math> than is <math>f(x_0)</math>.

1617-257/TUT-R-8

2016-11-07T20:32:47Z

Jeffim:

On 11/3/16, we discussed some questions from the exam:

'''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</math> has a limit point.

''Proof''. If <math>E</math> has no limit points, then <math>E</math> is a closed subset of a compact space and is therefore compact in itself. Since each point of <math>E</math> is isolated, we may find for each point <math>e \in E</math> a neighborhood <math>U_e</math> such that <math> E \cap U_e = \{ e \}</math>. The collection <math>\{ U_e\}_{e \in E}</math> is an open cover of E which clearly has no finite subcover.

'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.

''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is closer to <math>z</math> than is <math>f(x_0)</math>.

1617-257/TUT-R-8

2016-11-07T20:32:24Z

Jeffim:

On 11/3/16, we discussed some questions from the exam:

'''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</math> has a limit point.

''Proof''. If <math>E</math> has no limit points, then <math>E</math> is a closed subset of a compact space and is therefore compact in itself. Since each point of <math>E</math> is isolated, we may find for each point <math>e \in E</math> a neighborhood <math>U_e</math> such that <math> E \cap U_e = \{ e \}</math>. The collection <math>\{ U_e\}_{e \in E}</math> is an open cover of E which clearly has no finite subcover.

'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.

''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is strictly closer to <math>z</math> than is <math>f(x_0)</math>.

1617-257/TUT-R-8

2016-11-07T20:31:35Z

Jeffim: Created page with "On 11/3/16, we discussed some questions from the exam: '''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</..."

On 11/3/16, we discussed some questions from the exam:

'''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</math> has a limit point.

''Proof''. If <math>E</math> has no limit points, then <math>E</math> is a closed subset of a compact space and is therefore compact in itself. Since each point of <math>E</math> is isolated, we may find for each point <math>e \in E</math> a neighborhood <math>U_e</math> such that <math> E \cap U_e = \{ e \}</math>. The collection <math>\{ U_e\}_{e \in E}</math> is an open cover of E which clearly has no finite subcover.

'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.

''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>.

1617-257

2016-11-07T20:12:30Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

* [[1617-257/TUT-R-8| Tutorial R-8 comments]]

1617-257

2016-11-01T21:50:17Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

1617-257/HW-5

2016-11-01T21:49:58Z

Jeffim: Blanked the page

1617-257/HW-5

2016-11-01T21:46:50Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

----

'''Comments'''

'''Z. graded 8-1'''.

'''I. graded problem 8-5'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). One way to proceed is to show that the function in question is locally injective which then allows one to employ the IFT locally.

1617-257/HW-5

2016-11-01T21:46:25Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

----

'''Comments'''

'''Z. graded 8-1'''.

'''I. graded problem 8-5'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). One way to proceed is to show that the function is locally injective so that one can employ the IFT locally.

1617-257/HW-5

2016-11-01T21:31:52Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

----

'''Comments'''

'''Z. graded 8-1'''.

'''I. graded problem 8-5'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). Among other things, one could have argued that the function is locally injective so that one can employ the IFT locally.

1617-257/HW-5

2016-11-01T21:31:08Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

----

'''Comments'''

'''Z. graded 8-1'''.

'''I. graded problem 8-5'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). Among other things, one could have either delved into a proof of the IFT or argued that the function is locally injective so that one can employ the IFT locally.

1617-257/HW-5

2016-11-01T21:19:26Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

----

'''Comments'''

'''J.A.Z. graded problem 1 from section 8'''.

'''J.I. graded problem 5 from section 8'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). Among other things, one could have either delved into a proof of the IFT or argued that the function is locally injective so that one can employ the IFT locally.

1617-257/HW-5

2016-11-01T21:17:38Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

---

'''Comments'''

'''J.A.Z. graded problem 1 from section 8'''.

'''J.I. graded problem 5 from section 8'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). Among other things, one could have either delved into a proof of the IFT or argued that the function is locally injective so that one can employ the IFT locally.

1617-257/HW-5

2016-11-01T21:17:11Z

Jeffim:

'''General grading scheme'''

Two problems graded out of 15 points each.

'''J.A.Z. graded problem 1 from section 8'''.

'''J.I. graded problem 5 from section 8'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). Among other things, one could have either delved into a proof of the IFT or argued that the function is locally injective so that one can employ the IFT locally.

1617-257

2016-11-01T21:12:19Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/HW-5| HW-5 comments]]

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

1617-257/HW-5

2016-11-01T21:11:19Z

Jeffim:

'''J.I. graded problem 5 from section 8'''. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place). Among other things, one could have either delved into a proof of the IFT or argued that the function is locally injective so that one can employ the IFT locally.

1617-257/HW-5

2016-11-01T21:09:21Z

Jeffim:

1617-257/HW-5

2016-11-01T21:09:09Z

Jeffim: Created page with "J.I. graded problem 5 from section 8. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherw..."

J.I. graded problem 5 from section 8. Many employed the inverse function theorem (IFT) inappropriately here (injectivity is a requirement for the statement of the IFT - otherwise, the conclusion that the function in question is invertible doesn't make sense in the first place).

1617-257

2016-11-01T21:06:53Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

* [[1617-257/HW-5| HW-5 comments]]

1617-257/TUT-R-7

2016-10-29T20:42:12Z

Jeffim: Created page with "On 10/28/16, we discussed the following problem: Suppose <math>F : \mathbb{R}^2 \to \mathbb{R}^2</math> is <math>C^1</math> with <math>F(t^4, e^{t^2}) = (4,5)</math> for all ..."

On 10/28/16, we discussed the following problem:

Suppose <math>F : \mathbb{R}^2 \to \mathbb{R}^2</math> is <math>C^1</math> with <math>F(t^4, e^{t^2}) = (4,5)</math> for all <math>t \in \mathbb{R}</math>.
Prove that <math>DF(0,1)</math> is not invertible.

----

Let <math>\gamma(t) := (t^4, e^{t^2})</math>. At the end of the tutorial, a couple of students pointed out that it's also true that <math>DF(\gamma(t))</math> is not invertible for any <math>t \in \mathbb{R}</math>. It's easy to check that <math>\gamma'(t) \neq 0</math> when <math>t \neq 0</math> and that <math>\gamma'(t)</math> is in the kernel of <math>DF(\gamma(t))</math>.

It takes some more care to prove what the problem is asking because one can't immediately deduce that <math>DF(\gamma(t))</math> has non-trivial kernel with <math>\gamma'(0) = 0</math>. A key observation here is that since <math>F</math> is <math>C^1</math>, invertibility of <math>DF</math> at <math>\gamma(0)</math> implies invertibility of <math>DF</math> at a neighborhood of <math>\gamma(0)</math>.

1617-257

2016-10-29T20:30:45Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

* [[1617-257/TUT-R-7| Tutorial R-7 comments]]

1617-257/TUT-R-6

2016-10-21T17:25:04Z

Jeffim:

On 10/20/16, we discussed how one can calculate the derivative of the inverse of the map <math>f: (0,1/2) \times (0, \pi/4) \to U \subset \mathbb R^2</math> where <math>f(r, \theta) := (r \cos \theta, r \sin \theta)</math>:

Method 1. Find <math>f^{-1}</math> explicitly and differentiate it.

Method 2. Use the inverse function theorem.

----

Students who did both calculations noted that Method 2 was simpler than Method 1.

1617-257/TUT-R-6

2016-10-21T17:24:42Z

Jeffim:

1617-257/TUT-R-6

2016-10-21T17:23:03Z

Jeffim:

On 10/20/16, we discussed how one can calculate the derivative of the inverse of a map <math>f: (0,1/2) \times (0, \pi/4) \to U \subset \mathbb R^2</math> where <math>f(r, \theta) := (r \cos \theta, r \sin \theta)</math>:

Method 1. Find <math>f^{-1}</math> explicitly and differentiate it.

Method 2. Use the inverse function theorem.

1617-257/TUT-R-6

2016-10-21T17:22:32Z

Jeffim: Created page with "On 10/20/16, we discussed how one can calculate the derivative of the inverse of a map <math>f: (0,1/2) \times (0, \pi/4) \to U \subset \mathbb R^2</math> where <math>f(r, \th..."

On 10/20/16, we discussed how one can calculate the derivative of the inverse of a map <math>f: (0,1/2) \times (0, \pi/4) \to U \subset \mathbb R^2</math> where <math>f(r, \theta) := (r \cos \theta, r \sin \theta)</math>:

Method 1. Find <math>f^{-1}</math> explicitly and differentiate it.

Method 2. Use the inverse function theorem.

1617-257

2016-10-21T17:17:24Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

* [[1617-257/TUT-R-6| Tutorial R-6 comments]]

1617-257/TUT-R-5

2016-10-14T15:34:01Z

Jeffim:

On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and
if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all <math>x \in U</math>
then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>.

We also proved the analogous statement if <math>f</math> is Lipschitz continuous instead of
having uniformly bounded derivative.

Lastly, we created a formulation for the problem if <math>U</math> is star-shaped rather
than convex.

1617-257/TUT-R-5

2016-10-14T15:32:16Z

Jeffim: Created page with "On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D..."

On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and
if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all <nowiki>x \in U</nowiki>,
then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>.

We also proved the analogous statement if <math>f</math> is Lipschitz continuous instead of
having uniformly bounded derivative.

Lastly, we created a formulation for the problem if <math>U</math> is star-shaped rather
than convex.

1617-257/TUT-R-4

2016-10-14T15:27:52Z

Jeffim:

On 10/6/16, we showed that <math>\mathbb R^\infty</math> is a metric space where compactness
is not equivalent to being closed and bounded. We also showed that <math>\mathbb
R^\infty</math> is not complete with respect to the norm we gave it.

----

We also approximated the values of <math>\sqrt{4.002}</math> and <math>2^{4.002}</math> with first
order derivatives.

1617-257/TUT-R-4

2016-10-14T15:27:37Z

Jeffim: Created page with "On 10/6/16, we showed that <math>\mathbb R^\infty</math> is a metric space where compactness is not equivalent to being closed and bounded. We also showed that <math>\mathbb R..."

On 10/6/16, we showed that <math>\mathbb R^\infty</math> is a metric space where compactness
is not equivalent to being closed and bounded. We also showed that <math>\mathbb
R^\infty</math> is not complete with respect to the norm we gave it.

---

We also approximated the values of <math>\sqrt{4.002}</math> and <math>2^{4.002}</math> with first
order derivatives.

1617-257

2016-10-14T15:16:57Z

Jeffim:

__NOEDITSECTION__
__NOTOC__
{{1617-257/Navigation}}
==Analysis II==
===Department of Mathematics, University of Toronto, Fall 2016 - Spring 2017===

{{1617-257/Crucial Information}}

===Text===
Our main text book will be ''Analysis on Manifolds'' by James R. Munkres; it is a required reading.

===Further Resources===

* [http://www.math.toronto.edu/cms/undergraduate-program/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://www.math.utoronto.ca/almut/MAT257.html 2007-08 class], by Almut Burchard.

* [http://www.math.toronto.edu/vtk/257Fall2010/ 2010-11 class], by Vitali Kapovitch.

* [http://www.math.utoronto.ca/rjerrard/257/syllabus.html 2011-12 class], by Robert Jerrard.

* My {{Pensieve Link|Classes/1617-257-AnalysisII/|1617-257 notebook}}.

{{1617-257:Dror/Students Divider}}

* [[1617-257/TUT-R-1| Tutorial R-1 comments]]

* [[1617-257/TUT-R-2| Tutorial R-2 comments]]

* [[1617-257/TUT-R-3| Tutorial R-3 comments]]

* [[1617-257/TUT-R-4| Tutorial R-4 comments]]

* [[1617-257/TUT-R-5| Tutorial R-5 comments]]

1617-257/Class Photo

2016-10-01T01:53:50Z

Jeffim: /* Who We Are... */

Our class on September 30, 2016:

[[Image:1617-257-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{1617-257/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=1

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}



{{Photo Entry|last=Im|first=Jeffrey|userid=jeffim|email=jim@ math. toronto. edu|location=bottom-right corner|comments=One of two TAs for this class. Student of George Elliott. Interested in C*-algebras/K-theory.}}

{{Photo Entry|last=Xu|first=Xinyi|userid=Maggie|email=xinyi. xu97@ gmail. com|location=fourth row, first girl from right,with red glasses|comments=Interested in modern Origami.}}




|}

1617-257/Class Photo

2016-10-01T01:53:31Z

Jeffim: /* Who We Are... */

Our class on September 30, 2016:

[[Image:1617-257-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{1617-257/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=1

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}



{{Photo Entry|last=Im|first=Jeffrey|userid=jeffim|email=jim@ math. toronto. edu|location=bottom-right corner|comments=One of two TAs for this class. Student of George Elliott. Interested in C*-algebras.}}

{{Photo Entry|last=Xu|first=Xinyi|userid=Maggie|email=xinyi. xu97@ gmail. com|location=fourth row, first girl from right,with red glasses|comments=Interested in modern Origami.}}




|}

1617-257/Class Photo

2016-10-01T01:52:15Z

Jeffim:

Our class on September 30, 2016:

[[Image:1617-257-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{1617-257/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=1

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}



{{Photo Entry|last=Im|first=Jeffrey|userid=jeffim|email=jim@ math. toronto. edu|location=bottom-right corner|comments=One of two TAs for 257.}}

{{Photo Entry|last=Xu|first=Xinyi|userid=Maggie|email=xinyi. xu97@ gmail. com|location=fourth row, first girl from right,with red glasses|comments=Interested in modern Origami.}}




|}

1617-257/TUT-R-3

2016-09-30T16:48:32Z

Jeffim:

On 9/29/16, we discussed three notions of compactness in <math>\mathbb{R}^n</math> equipped with the usual topology:

(1) closed and bounded

(2) subsequential compactness

(3) every open cover admits a finite subcover

We will tacitly assume that this is the topology we're giving <math>\mathbb{R}^n</math> for the remainder of this post.

----

* We proved that (1) and (2) are equivalent.

* Statements (2) and (3) are equivalent in general metric spaces.

* (1) is not necessarily equivalent to (2) or (3) in other settings (and even non-contrived settings: that is, settings which are not just around for the sake of counterexample. There is an abundance of examples arising from basic objects of study in functional analysis.).

1617-257/TUT-R-3

2016-09-30T16:47:38Z

Jeffim:

On 9/29/16, we discussed three notions of compactness in <math>\mathbb{R}^n</math> equipped with the usual topology:

(1) closed and bounded

(2) subsequential compactness

(3) every open cover admits a finite subcover

We will tacitly assume that this is the topology we're giving <math>\mathbb{R}^n</math> for the remainder of this post.

----

* We proved that (1) and (2) are equivalent.

* Statements (2) and (3) are equivalent in general metric spaces.

* (1) is not necessarily equivalent to (2) or (3) in other settings (and even non-contrived settings. That is, settings which are not just produced for the sake of counterexample. There is an abundance of examples arising from basic objects of study in functional analysis.).