References for Linear Algebra (on infinite dimensional vector spaces) preferably with an eye on Functional Analysis

Question

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.

Answer 1

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).