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Are infinite-dimensional vector spaces systematically and rigorously discussed in any linear algebra text(s)?

If so, please give a few here, as well as any courses with online content taught using that (those) text(s).

Saaqib Mahmood
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    Infinite dimensional vector spaces are basically function spaces, so you might prefer a more advanced topology book, or an intro to functional analysis. Once the unit ball is no longer compact, everything changes. –  May 12 '20 at 16:06
  • The problem is that most of the statements in linear algebra require a finite dimensional vector space. And there are some concepts, for example the matrix representation of a linear map with respect to some bases, you don't have in a vector space of infinite dimension. – Fakemistake May 12 '20 at 17:50
  • See linear algebra in infinite dimension, especially the reference in the answer (FYI, my go-to book for this topic) and the references found by following the links in the comments. – Dave L. Renfro May 12 '20 at 19:12

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They're often covered in textbooks on functional analysis with added structure, i.e. a norm, completeness, or an inner product. You could try Rudin's Real and Complex Analysis for the basics of infinite dimensional vector spaces and Conway's A Course on Functional Analysis covers more.

Linear algebra textbooks don't typically discuss much of the infinite dimensional case, but many of the basics of linear algebra extend to the infinite dimensional case pretty straightforwardly, e.g. inner product spaces have basically the same proofs in finite or infinite dimensions.

William Bell
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