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nLab > Latest Changes: Stone-Weierstrass theorem

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeApr 4th 2025
- (edited Apr 4th 2025)
- PermaLink
Author: Urs
Format: MarkdownItexIt is being pointed out to me by email that this entry says about the algebra $C_0(X)$ of functions vanishing at infinity that: > $C_0(X)$ is no longer a Banach space (due to [revision 1](https://ncatlab.org/nlab/revision/Stone-Weierstrass+theorem/1) by Todd Trimble, way back in October 2009) This seems odd, as $C_0(X)$ is a standard example of a Banach space, unless something else is meant here. It seems nothing in the entry depends on this side-remark, so that it may be worth deleting. <a href="https://ncatlab.org/nlab/revision/diff/Stone-Weierstrass+theorem/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone-Weierstrass+theorem/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Stone-Weierstrass+theorem">current</a>

It is being pointed out to me by email that this entry says about the algebra $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math>$ of functions vanishing at infinity that:

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math>$ is no longer a Banach space

(due to revision 1 by Todd Trimble, way back in October 2009)

This seems odd, as $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math>$ is a standard example of a Banach space, unless something else is meant here.

It seems nothing in the entry depends on this side-remark, so that it may be worth deleting.

diff, v11, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeApr 8th 2025
- PermaLink
Author: Urs
Format: MarkdownItexHave deleted the parenthesis starting by saying that $C_0(X)$ is not a Banach space. <a href="https://ncatlab.org/nlab/revision/diff/Stone-Weierstrass+theorem/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone-Weierstrass+theorem/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Stone-Weierstrass+theorem">current</a>

Have deleted the parenthesis starting by saying that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math>$ is not a Banach space.

diff, v12, current

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nLab > Latest Changes: Stone-Weierstrass theorem