What is a Hilbert space?

Question

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?

I looked at the Wikipedia article but was left bemused by "A vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space." — Sachin Kainth, Jun 10 '13 at 16:05
The article is referring to a Fourier series expansion: http://en.wikipedia.org/wiki/Fourier_series . Basically, you can write any periodic function (such as one vibration of a string) as a sum of sines and cosines. This makes it easier to work with. The set of all such expansions forms a "Hilbert space": you can add them, multiply them by constants, etc. — Potato, Jun 10 '13 at 16:13
No I don't to be honest. I think a vector is a two dimensional list of numbers. — Sachin Kainth, Jun 10 '13 at 16:22
Probably not a topic for a layman (i.e., someone who has no knowledge of "vector spaces"). — GEdgar, Jun 10 '13 at 21:05

score 8 · Accepted Answer · answered Jun 10 '13 at 16:44

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):

cat

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 $0$.

score 6 · Answer 2 · answered Jun 10 '13 at 16:19

6

A Hilbert space is a complete inner product space.

answered Jun 10 '13 at 16:19

anegligibleperson

2,509

2

... and it is an abstract vector space of course. – al-Hwarizmi Jun 10 '13 at 20:31

score 3 · Answer 3 · answered Jun 10 '13 at 19:49

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.

What is a Hilbert space?

3 Answers3