# What is a Hilbert space?

##### 1 Answer

Hilbert space is a set of elements with certain properties, namely:

it's a vector space (so, there are operations on its elements typical for vectors, like multiplication by a real number and addition that satisfy commutative and associative laws);

there is a scalar (sometimes called inner or dot) product between any two elements that results in a real number.

For example, our three-dimensional Euclidean space is an example of a Hilbert space with scalar product of

More interesting example is a space of all continuous functions on a segment

In quantum physics Hilbert space plays a very important role as a function that describes the state of a system

I can recommend

http://www.phy.ohiou.edu/~elster/lectures/qm1_1p2.pdf

as an introduction into usage of Hilbert space in quantum physics.