# What is a Hilbert space?

Sep 9, 2014

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 $x = \left({x}_{1} , {x}_{2} , {x}_{3}\right)$ and $y = \left({y}_{1} , {y}_{2} , {y}_{3}\right)$ equal to $\left(x , y\right) = {x}_{1} \cdot {y}_{1} + {x}_{2} \cdot {y}_{2} + {x}_{3} \cdot {y}_{3}$.

More interesting example is a space of all continuous functions on a segment $\left[a , b\right]$ with a scalar product defined as
$\left(f , g\right) = {\int}_{a}^{b} \left[f \left(x\right) \cdot g \left(x\right)\right] \mathrm{dx}$

In quantum physics Hilbert space plays a very important role as a function that describes the state of a system $\Psi$ is an element of a Hilbert space.

I can recommend
http://www.phy.ohiou.edu/~elster/lectures/qm1_1p2.pdf
as an introduction into usage of Hilbert space in quantum physics.