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Hilbert space (Definition)

Basic concepts

Definition 1.1   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 $v \in V, \langle{v, v \rangle} \geqslant 0$.
  • (2) If $\langle{v, v \rangle} = 0$ then $v = 0$.
  • (3) For all $v$ and $w$ in $V$,

    $\displaystyle \langle{v, w \rangle} = \overline{\langle{w,v \rangle}}$
    .
  • (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

    $\displaystyle \langle{av,w \rangle}= a \langle{v,w \rangle}$
    .(The inner product is linear in the first variable, and conjugate linear in the second.)
Definition 1.2   A Banach space $(X,\left\Vert{\cdot}\right\Vert)$ is a normed vector space such that $X$ is complete under the metric induced by the norm $\left\Vert{\cdot}\right\Vert$.

Hilbert space

Definition 2.1   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 $\left\Vert{v_m} - {v_n}\right\Vert \to 0$ as $m$ and $n$ both tend to infinity, there is in $V$, a vector $v_{\omega} \in V$ such that $\left\Vert{v_m} - {v_{\omega}}\right\Vert \to 0$ as $n \to \infty$. (In quantum physics, all Hilbert spaces are tacitly assumed to be infinite dimensional)

Remarks

Sequences with the property that $lim _{m \to \infty, n \to \infty} \left\Vert{v_m} - {v_n}\right\Vert = 0$ 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.



"Hilbert space" is owned by bci1.
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See Also: quantum groups and von Neumann algebras, topic on algebraic foundations of quantum algebraic topology, Poisson ring, vector space

Also defines:  Cauchy sequence, vector space, conjugate linear, inner product, norm, Banach space, metric space, metric induced norm, norm induced by inner product
Keywords:  Hilbert space, quantum state space, norm, Banach space, vector space, Cauchy sequence, inner product, norm, Banach space, metric space, metric induced norm, norm induced by inner product

Attachments:
wave function space (Topic) by bloftin

Cross-references: quantum mechanics, works, vectors, metric, function, scalars
There are 62 references to this object.

This is version 38 of Hilbert space, born on 2009-05-21, modified 2009-05-21.
Object id is 768, canonical name is HilbertSpace3.
Accessed 5056 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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