Hopefully easy question about sups of continuous functions

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Discussion Overview

The discussion revolves around the properties of continuous functionals on normed spaces, specifically examining the relationship between the supremum of a functional over the interior of the unit ball and the supremum over the boundary of the unit ball. Participants explore whether the equality holds and seek clarification on definitions and examples.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the equality \(\sup_{\|x\| < 1} |f(x)| = \sup_{\|x\| = 1} |f(x)|\) holds for continuous functionals on normed spaces and asks for a counterexample if it does not.
  • Another participant notes that a normed space must be a vector space but questions whether a vector space has to be connected.
  • A suggestion is made to consider the case of the real numbers \(\mathbb{R}\) and whether there exists a function that achieves a larger value within the unit interval than at the endpoints 1 and -1, indicating a potential misunderstanding of the term "functional" as possibly implying linearity.
  • A later reply reiterates the previous point about the real numbers and expresses uncertainty about the general case for arbitrary normed spaces, while clarifying that the term "functional" was intended to mean "linear functional."

Areas of Agreement / Disagreement

Participants express uncertainty regarding the general case for arbitrary normed spaces, and there is no consensus on whether the proposed equality holds. Multiple viewpoints are presented, particularly concerning the definitions and implications of "functional."

Contextual Notes

There is ambiguity regarding the definitions of "functional" and "linear functional," and the discussion does not resolve whether the equality holds in all cases or if counterexamples exist.

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If f is a continuous functional on a normed space, do you have

[tex] \sup_{\|x\| < 1} |f(x)| = \sup_{\|x\| = 1} |f(x)|[/tex]

If so, why? If not, can someone provide a counterexample?
 
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That's an interesting question. A normed space must be a vector space right? But does a vector space have to be connected?
 
try it on R. can you think of a function which gets larger somewhere inside the unit interval than it is at 1, -1? maybe you meant linear. or maybe functional means linear. then it seems true.
 
mathwonk said:
try it on R. can you think of a function which gets larger somewhere inside the unit interval than it is at 1, -1? maybe you meant linear. or maybe functional means linear. then it seems true.

Yeah, I did mean to put "linear functional" above...I guess I was under the impression that "functional" implies "linear functional", though I'm not at all sure that's the case.

And yeah, it clearly seems true on R, but for an arbitrary normed space, I'm not so sure...
 

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