1617-257/TUT-R-3 - Revision history

Jeffim at 16:48, 30 September 2016

2016-09-30T16:48:32Z

← Older revision		Revision as of 12:48, 30 September 2016
Line 16:		Line 16:
	* Statements (2) and (3) are equivalent in general metric spaces.		* 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.).		* (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.).

Jeffim at 16:47, 30 September 2016

2016-09-30T16:47:38Z

← Older revision		Revision as of 12:47, 30 September 2016
Line 16:		Line 16:
	* Statements (2) and (3) are equivalent in general metric spaces.		* 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 examples arising from basic objects of study in functional analysis.).		* (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.).

Jeffim at 16:35, 30 September 2016

2016-09-30T16:35:26Z

← Older revision		Revision as of 12:35, 30 September 2016
Line 16:		Line 16:
	* Statements (2) and (3) are equivalent in general metric spaces.		* Statements (2) and (3) are equivalent in general metric spaces.

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

Jeffim at 16:34, 30 September 2016

2016-09-30T16:34:18Z

← Older revision		Revision as of 12:34, 30 September 2016
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	----		----

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

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

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

Jeffim at 16:33, 30 September 2016

2016-09-30T16:33:33Z

← Older revision		Revision as of 12:33, 30 September 2016
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	We will tacitly assume that this is the topology we're giving <math>\mathbb{R}^n</math> for the remainder of this post.		We will tacitly assume that this is the topology we're giving <math>\mathbb{R}^n</math> for the remainder of this post.


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

Jeffim: Created page with "On 9/29/16, we discussed three notions of compactness in $\mathbb{R}^n$ equipped with the usual topology: (1) closed and bounded (2) subsequential compactness (3..."

2016-09-30T16:33:11Z

Created page with "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..."

New page

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 non-contrived settings (i.e. Settings which are not just produced for the sake of counterexample. There is an abundance examples arising from basic objects of study in functional analysis.).