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i am mostly translating my question (from french) so sorry for the rough english, but i would like to know some examples of properties that a finite-dimensional vector space has but an infinite-dimensional vector space doesn't.

realbluebells
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  • A finite-dimensional vector space $V$ is always isomorphic to its dual space, which is not true for infinite dimensional vector spaces – Alessandro May 19 '21 at 20:26
  • Separability, Completeness, Heine-Borel, All linear operators between finite-dimensional normed spaces are bounded, all norms are equivalent. –  May 19 '21 at 20:28
  • @user That's more like for topological vector spaces, and the OP seems to want something only from linear algebra, without topology... – DonAntonio May 19 '21 at 20:29
  • https://www.reddit.com/r/math/comments/bdj27x/linear_algebra_theorems_that_dont_generalize_to/ has lots of examples – Rhys Steele May 19 '21 at 20:30
  • if $V$ is finite dimensional, then a linear map with a one sided inverse from $V$ to itself must be invertible, but this is false for infinite dimensional spaces. – lulu May 19 '21 at 20:36
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    See this – Jean Marie May 19 '21 at 20:52

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Here is one example. If $V$ is a vector space and $f:V\rightarrow V$ is linear and injective, it is a priori surjective if $V$ is finite dimensional. This fails in the infinite dimensional case. Similarly, if $f$ is surjective, it must also be injective if $V$ is finite dimensional. This fails in the infinite dimensional case.

ncmathsadist
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