Hilbert Space

Hilbert space is a special kind of vector space satisfying the following properties in addition to the properties of a vector space:

• Has an inner product operation:\[<\psi_1, \psi_2> \in \mathbb{C} \]
  • Conjugate symmetry: \[<\psi_1, \psi_2> = <\psi_2, \psi_1>^*\]
  • Linearity wrt. 2nd vector: \[<\psi_1, a\psi_2 + b\psi_3> = a<\psi_1, \psi_2> + b<\psi_1, \psi_3>\]
  • Antilinear wrt. 1st vector: \[<a\psi_1 + b\psi_2, \psi_3> = a^{*}<\psi_1, \psi_3> + b^{*}<\psi_2, \psi_3>\]
  • Positive definite: \[<\psi, \psi> = |\psi|^2\]
• Hilbert spaces are separable, i.e. they contain a countable, dense subset. (explanation after the properties)
• Hilbert Spaces are complete.

Dense sets

A subset \(X\) of set \(A\) is called dense when every element in \(A\) is either in \(X\) or is a limit point of an element in \(X\).

For example, the set of rational numbers \(\mathbb{Q}\) is a dense subset of the set of real numbers \(\mathbb{R}\)

Complete sets

A set \(A\) is said to be complete if all the cauchy sequences in \(A\) converge to some element in \(A\). For example, \(\mathbb{Q}\) is not a complete set because the cauchy sequence \(a_0 = 1, a_n = \frac{a_{n-1}}{2}+\frac{1}{a_n}\) converges to \(\sqrt{2}\).