I am taking a self study graduate course on Functional Analysis and the notes are quite terse. I am struggling in particular with the linear algebra background part, including dealing with infinite dimensional vector spaces.

So I was looking for a good reference to complement the notes. It would be particularly good if the book involved solved exercises and detailed proofs.

Thank you.

  • $\begingroup$ Any good, undergraduate level introductions to Functional Analysis? might not be a duplicate, but might still be helpful. $\endgroup$ – Mark S. Oct 28 '20 at 13:08
  • $\begingroup$ @Mark Thanks for your input but I already saw thst wuestion amd run into the same problems than with other references: it covers topology and banach and hilbert spaces but not arbitrary infinite dimensional vector spaces and isomorphisms etc... But thank you anyways. $\endgroup$ – Student Oct 28 '20 at 14:20

I'm not aware of any such reference that isn't either a text on functional analysis or a text on commutative algebra. It would help if you were more specific about what you wanted to know about infinite-dimensional vector spaces. Most of what is worth knowing about them is a list of things that don't generalize from finite-dimensional vector spaces. If $V$ is infinite-dimensional:

  • In general you can't define the determinant, trace, or characteristic polynomial of an endomorphism $T : V \to V$.
  • It's not true that $T : V \to V$ must have an eigenvalue even over an algebraically closed field.
  • It's not true that $V$ has the same dimension as $V^{\ast}$.
  • It's not true that an injection $V \to V$ is automatically a bijection, and similarly for surjections.

Then there's some stuff that relies on the axiom of choice:

  • $V$ still has a basis, but only if you assume the axiom of choice (and in fact it's a famous result due to Blass that AC is equivalent to the assertion that every vector space has a basis). Due to the independence of AC it follows that it's consistent with ZF that there are infinite-dimensional vector spaces without bases.
  • Linear functionals still separate points, but again only if you assume the axiom of choice. It's consistent with ZF that there are infinite-dimensional vector spaces with no nonzero linear functionals (see this MO thread).
  • $\begingroup$ Thank you. I am not looking for commutative algebra. Do you have any refereces of texts on functional analysis that treat infinite dimensional vector spaces in generality? For example, I would like to see a treatment of basis for infinite dimensional spaces witb maybe some examples and I would also like to see the proof for infinite dimensional spaces rhat two vector spaces are isomorphic if and only if any of their basis has the same cardinality, a proof that the basis for thr dual space of V has cardinality 2^card(B) for a basis B of V. This sort of thing. $\endgroup$ – Student Oct 29 '20 at 22:26
  • $\begingroup$ @Student: I'm not aware of a reference for any of that. I would consider all of those basically annoying exercises. Many infinite-dimensional vector spaces don't have explicit bases in any reasonable sense so it's unclear what "some examples" would mean here. $\endgroup$ – Qiaochu Yuan Oct 29 '20 at 22:36
  • $\begingroup$ The way the proofs go for bases in the presence of choice is a Zorn's lemma argument: you start with a linearly independent set of vectors and just keep adding more until you get a maximal linearly independent set, which must be a basis. For a proof that two bases have the same size see Wikipedia: en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces#Proof $\endgroup$ – Qiaochu Yuan Oct 29 '20 at 22:39

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