Is R^w a first category topological vector space?

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Discussion Overview

The discussion revolves around the classification of the topological vector space R^w in relation to first category spaces. Participants explore the implications of R^w being expressed as a union of finite-dimensional subspaces and the conditions under which it may or may not satisfy the properties of first category spaces.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant references a problem from Rudin's Functional Analysis, questioning whether R^w can be considered a first category space if it is the union of countably many finite-dimensional subspaces.
  • Another participant challenges the claim that R^w can be represented as the union of R^n for n going from 1 to infinity, suggesting that this assertion is incorrect.
  • A participant reflects on the nature of the union of finite-dimensional spaces, expressing confusion about whether the construction truly represents a union of finite-dimensional sets, given the infinite-dimensional nature of the total union.
  • There is a query about the proof of "nonempty interior" in relation to R^w, indicating a need for clarification on this aspect.
  • One participant seeks to understand if the previous comments imply that R^w is indeed a first category space, while also noting that the closure of open balls in R^w has non-empty interior.

Areas of Agreement / Disagreement

Participants express differing views on the classification of R^w, with some asserting it cannot be a first category space while others question the validity of the union argument. The discussion remains unresolved with competing perspectives on the nature of R^w and its dimensional properties.

Contextual Notes

There are unresolved assumptions regarding the definitions of first category spaces and the properties of the union of finite-dimensional spaces. The implications of non-empty interior in R^w also remain to be clarified.

redrzewski
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This is from Rudin, Functional Analysis 2.1. Not homework.

If X is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove X is first category in itself.

What about this example? Take R^n (standard n-dimensional space of reals) as each of the finite-dimensional subspaces. Then the union as n goes from 1 to infinity will be R^w.

R^w is infinite-dimensional, and it will contain closed sets that have non-empty interior. R^w seems like it will satify the axioms of the topological vector space. Hence this would contradict the problem.

What am I missing?

thanks
 
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redrzewski said:
Then the union as n goes from 1 to infinity will be R^w.

This is not true, just as R \cup R \neq R^2.
 
I was assuming R U R^2 U R^3 = R^3, etc.

I'm getting hung up on if this construction actually is a union of finite dimensional sets. On the one hand, given any set in the union, it has finite dimension.

On the other hand, since every set is a superset of the ones of lower dimensions, and that the total union has infinite dimension, it seems like one of the sets in the union must have infinite dimension.
 
Last edited:
redrzewski said:
What am I missing?

the proof of "nonempty interior"
 
I don't understand the comment. Are you saying that R^w is 1st category, and that I need to prove it?

But the closure of any open ball B(0,r) in a metric for R^w has non-empty interior, right?
 

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