Skip to main content
MathOverflow

Return to Question

replaced http://mathoverflow.net/ with https://mathoverflow.net/
Source Link
Community Bot
Community Bot
  • 1
  • 2
  • 3

A popular pair of exercises in first courses on functional analysis prove the following theorem:

The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.

My question is, is the "only if" part of this (i.e., that the unit ball of an infinite-dimensional Banach space is noncompact) necessarily true without some form of the axiom of choice? The usual proof uses the Hahn--Banach theorem, which may reasonably be regarded as a weak form of the axiom of choice (see this answerthis answer, and other answers to the same question, for some interesting points related to this).

A popular pair of exercises in first courses on functional analysis prove the following theorem:

The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.

My question is, is the "only if" part of this (i.e., that the unit ball of an infinite-dimensional Banach space is noncompact) necessarily true without some form of the axiom of choice? The usual proof uses the Hahn--Banach theorem, which may reasonably be regarded as a weak form of the axiom of choice (see this answer, and other answers to the same question, for some interesting points related to this).

A popular pair of exercises in first courses on functional analysis prove the following theorem:

The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.

My question is, is the "only if" part of this (i.e., that the unit ball of an infinite-dimensional Banach space is noncompact) necessarily true without some form of the axiom of choice? The usual proof uses the Hahn--Banach theorem, which may reasonably be regarded as a weak form of the axiom of choice (see this answer, and other answers to the same question, for some interesting points related to this).

Source Link
Mark Meckes
Mark Meckes
  • 11.4k
  • 3
  • 38
  • 69

Is there an infinite-dimensional Banach space with a compact unit ball?

A popular pair of exercises in first courses on functional analysis prove the following theorem:

The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.

My question is, is the "only if" part of this (i.e., that the unit ball of an infinite-dimensional Banach space is noncompact) necessarily true without some form of the axiom of choice? The usual proof uses the Hahn--Banach theorem, which may reasonably be regarded as a weak form of the axiom of choice (see this answer, and other answers to the same question, for some interesting points related to this).