Although it is about a specific application, optimization, it is a good book to get a sense of infinite dimensional vector spaces. I would also recommend Halmos. His book surreptitiously introduces you to that subset of linear algebraic notions that survive inti infinite dimensional spaces.
if you take the spectral theorem, for example, there is a direct connection between linear algebra and functional analysis, basically it's linear algebra in infinite dimensions
https://ia801706.us.archive.org/7/items/in.ernet.dli.2015.14...
Luenberger, Optimization By Vector Space Methods.
Although it is about a specific application, optimization, it is a good book to get a sense of infinite dimensional vector spaces. I would also recommend Halmos. His book surreptitiously introduces you to that subset of linear algebraic notions that survive inti infinite dimensional spaces.