What is a Hilbert space? 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?
 A: For you are a programmer, here is an example involving $1$'s and $0$'s.
Schrödinger's cat is argubably the most widespread "hard" science thought experiment that invaded the pop culture most.
The state of the cat lives in a Hilbert space. The following comic is pretty illustrative (from abstrusegoose):

In this comic, several characteristics of Hilbert spaces are shown (not all though, for some mathematical facts are hard to interpret without rigorous formality). 


*

*Vector in a vector space does not have to be two dimensional list of numbers: "Vector" can be a very abstract function $\psi(x)$. Here it is the state of the cat, namely $|0\rangle$ (dead) and $|1\rangle$ (alive), later the author added a new discovered basis in this space $|\mathrm{LOL}\rangle$.

*Linearity: The member of a Hilbert spaces can be linearly combined with another member in this Hilbert spaces:
$$
|\psi\rangle  = \frac{1}{\sqrt{2}}|0\rangle +  \frac{1}{\sqrt{2}}|1\rangle,
$$
or rather for $\alpha^2+\beta^2 = 1$:
$$
|\psi\rangle  = \alpha|0\rangle +  \beta|1\rangle.
$$
We can get members in the same Hilbert space in a more abstract way.

*Inner product structure: where this "inner product" can be view more abstractly as well other than $a\cdot b$. In this case, it can be interpreted as observation collapsing the states:  $0$ state inner product with an arbitrary state $\psi$
$$
\langle 0 | \psi \rangle = \alpha \langle 0 |0\rangle = \alpha .
$$
Once we observe the cat' status, the probability we observe the cat in $|0\rangle $ is $\alpha^2$. Inner product gives us the square root of the probability that we observe an arbitrary state $\psi$ in a fixed known state $0$.
A: A Hilbert space is a complete inner product space.
A: 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.
