I was reading this book, when the ideas of function spaces, functions as vectors, functions as elements of vector spaces, functional analysis clicked on me. I

I was reading this book, when the ideas of function spaces, functions as vectors, functions as elements of vector spaces, functional analysis clicked on me. I am not sure if this book is particularly good or better than other books. (Well, it still looks like a very gentle introduction to the topic.) But as per your question, this was the book at the right time for me. "Fourier Series and Orthogonal Functions" by Harry S. Davis. https://www.amazon.com/dp/0486659739/ https://www.amazon.com/dp/0486659739/