Scholarlink

In Theorem 1.5.4 we proved that the closed unit ball of an infinite-dimensional normed space is not compact. This strong contrast with finite dimension indicates that the existence of compact sets that are not contained in finite-dimensional subspaces is a delicate issue. Examples exist: we saw in Exercise 1.8.35 that the Hilbert cube is compact in \(\ell _2\) . The issue is not just the existence or non-existence of many compact sets; the central point is that those sets that we would like to be compact, namely the closed balls, are not.

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Weak Topologies

  • 章节

  • 发表日期：

    2025年

查看全文
  • Geraldo Botelho,
  • Daniel Pellegrino,
  • Eduardo Teixeira

摘要

In Theorem 1.5.4 we proved that the closed unit ball of an infinite-dimensional normed space is not compact. This strong contrast with finite dimension indicates that the existence of compact sets that are not contained in finite-dimensional subspaces is a delicate issue. Examples exist: we saw in Exercise 1.8.35 that the Hilbert cube is compact in \(\ell _2\) . The issue is not just the existence or non-existence of many compact sets; the central point is that those sets that we would like to be compact, namely the closed balls, are not.