TITLE: What is a Hilbert space?
QUESTION [8 upvotes]: I've just seen a question about Hilbert Subspaces.
This made me wonder what a Hilbert space is.  
Can anyone explain in layman's terms?

REPLY [3 votes]: Possibly the best explanation (and motivation for the subject) I know about that would be suitable for someone at the high school algebra to elementary calculus level are two chapters in the following book.
The Mathematical sciences: A Collection of Essays (1969). For the table of contents, see here or here.
See the chapters Functional Analysis by Jacob T. Schwartz (pp. 72-83) and Vector Spaces and Their Applications by Edward James McShane (pp. 84-96).
Also good is Chapter 19: Functional Analysis by Israel M. Gelfand in Mathematics: Its Content, Methods and Meaning. For the table of contents, see here.