Difference between revisions of "For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal"
From Maths
(Created page with "__TOC__ ==Statement== Let {{M|(X,\mathcal{J},}}\mathbb{K} }}{{M|)}} be a topological vector space and let {{M|(Y,\mathbb{K})}} be a vector sub...") |
(Adding picture of proof. Added link to A proper vector subspace of a topological vector space has no interior) |
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+ | : This is a [[precursor theorem]] to "''[[a proper vector subspace of a topological vector space has no interior]]''". | ||
__TOC__ | __TOC__ | ||
==Statement== | ==Statement== | ||
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** In words: if there exists a ''[[non-empty]]'' [[open set]] of {{M|(X,\mathcal{J})}}, say {{M|U}} | ** In words: if there exists a ''[[non-empty]]'' [[open set]] of {{M|(X,\mathcal{J})}}, say {{M|U}} | ||
==Proof== | ==Proof== | ||
− | {{Requires proof|grade= | + | {{Requires proof|grade=D|msg=Get a picture! |
+ | * Got a picture - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 17:51, 16 February 2017 (UTC) | ||
+ | <gallery> | ||
+ | File:ProofAProperVectorSubspaceHasNoInterior.JPG | The proof here does have precursors though | ||
+ | </gallery> | ||
+ | }} | ||
==See also== | ==See also== | ||
* {{XXX|Do this}} | * {{XXX|Do this}} |
Latest revision as of 17:51, 16 February 2017
- This is a precursor theorem to "a proper vector subspace of a topological vector space has no interior".
Contents
Statement
Let [ilmath](X,\mathcal{J},[/ilmath][ilmath]\mathbb{K} [/ilmath][ilmath])[/ilmath] be a topological vector space and let [ilmath](Y,\mathbb{K})[/ilmath] be a vector subspace of [ilmath](X,\mathbb{K})[/ilmath], then[1]:
- [ilmath](\exists U\in(\mathcal{J}-\{\emptyset\})[U\subseteq Y])\implies X\eq Y[/ilmath]
Proof
Grade: D
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
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The message provided is:
The message provided is:
See also
- TODO: Do this
References
Categories:
- Pages requiring proofs
- XXX Todo
- Theorems
- Theorems, lemmas and corollaries
- Functional Analysis Theorems
- Functional Analysis Theorems, lemmas and corollaries
- Functional Analysis
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