Limits and Supremum: Is It True?

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    Limits Supremum
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Discussion Overview

The discussion revolves around the convergence of a sequence in norm and its implications on the supremum of a set. Participants explore whether the supremum of the inner product remains finite under certain conditions, focusing on theoretical aspects of limits and supremum in mathematical analysis.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant presents a scenario involving a sequence ##a_n## converging in norm to ##a## and questions whether the supremum of the inner product with a set ##S## remains finite.
  • Another participant points out the need for proper formatting in LaTeX for clarity in mathematical expressions.
  • A correction is made regarding the formatting of LaTeX, along with a link to guidelines for using LaTeX on the forum.
  • A participant raises a question about the nature of the problem, suggesting it may be homework-related and emphasizes the need for showing work to receive hints. They introduce additional conditions involving a real number ##M## and an epsilon argument related to the convergence of the sequence.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original question regarding the supremum. There are multiple viewpoints, particularly regarding the nature of the problem and the conditions under which the supremum may be evaluated.

Contextual Notes

There are limitations in the discussion, including assumptions about the properties of the inner product and the set ##S## that are not fully explored. The mathematical steps and implications of the convergence are also not resolved.

Who May Find This Useful

This discussion may be useful for students and researchers interested in mathematical analysis, particularly those studying convergence, limits, and supremum in the context of functional analysis.

Mathvsphysics
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We have ##a_n## converges in norm to ##a## and a set ##S## such that for all ##n\ge 0##
$$\sup_{s\in S} <a_n,s><+\infty .$$ Is it true that ##\sup_{s\in S} <a,s><+\infty##
 
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You need to use two hashes (or two dollars) for Latex delimiters.
 
Is this a homework-type of problem? There is a format for those and you must show work before we can give hints.
Suppose ##M \in R## is such that ##sup<a_n,s> \lt M##. Also, suppose ##\epsilon \gt 0## and ##m\in N## are such that ##<a_n,a> \lt \epsilon## ##\forall n\gt m##. What can you say then?
 

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