PlanetPhysics/Hilbert Space 3

Basic concepts

An inner-product space with complex scalars, $\mathbf {C}$, is a vector space $V$ with complex scalars, together with a complex-valued function $\langle {v,w\rangle }$, called the inner product, defined on $V\times V$, which has the following properties:

• (1) For all $v\in V,\langle {v,v\rangle }\geq 0$.
• (2) If $\langle {v,v\rangle }=0$ then $v=0$.
• (3) For all $v$ and $w$ in $V$, $\langle {v,w\rangle }={\overline {\langle {w,v\rangle }}}$.
• (4) For all $v_{1},v_{2}$ and $w$ in $V$, $\langle {{v_{1}+v_{2}},w\rangle }=\langle {v_{1},w\rangle }+\langle {v_{2},w\rangle }$.
• (5) For all $v,w$ in V, and all scalars $a$, one has that $\langle {av,w\rangle }=a\langle {v,w\rangle }$.(The inner product is linear in the first variable, and conjugate linear in the second.)

A Banach space $(X,\left\|{\cdot }\right\|)$ is a normed vector space such that $X$ is complete under the metric induced by the norm $\left\|{\cdot }\right\|$.

Hilbert space

A Hilbert space is an inner product space which is complete as a metric space, that is for every sequence $\{v_{n}\}$ of vectors in $V$, if $\left\|{v_{m}}-{v_{n}}\right\|\to 0$ as $m$ and $n$ both tend to infinity, there is in $V$, a vector $v_{\omega }\in V$ such that $\left\|{v_{m}}-{v_{\omega }}\right\|\to 0$ as $n\to \infty$. (In quantum physics, all Hilbert spaces are tacitly assumed to be infinite dimensional)

Remarks

Sequences with the property that $lim_{m\to \infty ,n\to \infty }\left\|{v_{m}}-{v_{n}}\right\|=0$ are called Cauchy sequences . Usually one works with Hilbert spaces because one needs to have available such limits of Cauchy sequences. Finite dimensional inner product spaces are automatically Hilbert spaces. However, it is the infinite dimensional Hilbert spaces that are important for the proper foundation of quantum mechanics.

A Hilbert space is also a Banach space in the norm induced by the inner product, because both the norm and the inner product induce the same metric.