FDVS was written in 1942 when Halmos had just gotten his Ph.D. from J. Doob, author of
Stochastic Processes in my list, at U. Illinois, and was an assistant to John von Neumann at the Institute of Advanced Study in Princeton.
IIRC Hilbert space was a von Neumann idea: It is first, just a definition -- complete inner product (dot product in much of physics and engineering) space. But the good stuff is (1) importance of the examples and (2) the theorems that show the consequences, e.g., in Fourier theory.
Well, the vector spaces of most interest in linear algebra are actually (don't tell anyone) finite dimensional Hilbert spaces. So, one role of FDVS is to provide a text on linear algebra that is also an introduction to Hilbert space, that is, that tries to use ideas that work in any Hilbert space to get the basic results in linear algebra.
The treatment of self-adjoint transformations and spectral theory are likely the most influenced by this role.
This role is accomplished so well that sometimes physics students starting on quantum mechanics are advised to get at least the start they need on Hilbert space from FDVS.
Sure, a better start is the one chapter on Hilbert space in Rudin's Real and Complex Analysis. The chapter there on the Fourier transform is also good, short, all theorems nicely proved, the main, early results made clear.
Also a good start on the basic results of self-adjoint matrices are the inverse and implicit function theorems given as nice exercises in the third edition of Rudin's Principles .... And spectral theory is in Rudin's Functional Analysis. Also get a bonus of a nice treatment of distributions, that is, replace the Dirac delta function usage in quantum mechanics.
For how to get the eigen value and orthogonal eigen vector results for self-adjoint matrices from the inverse and implicit (these two go together like ice cream and cake) function theorems is in Fleming, Functions of Several Variables. Then you will be off and running on factor analysis, principle components, the polar decomposition, the singular value decomposition, and more.