## Hilbert space |

The **Hilbert space**, named after

Hilbert spaces arise naturally and frequently in *Hilbert space* for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the *the* Hilbert space.

- definition and illustration
- history
- examples
- applications
- properties
- operators on hilbert spaces
- constructions
- orthonormal bases
- orthogonal complements and projections
- spectral theory
- in popular culture
- see also
- remarks
- notes
- references
- external links

One of the most familiar examples of a Hilbert space is the ^{3}, and equipped with the **x** and **y**, and produces a real number **x** · **y**. If **x** and **y** are represented in

The dot product satisfies the properties:

- It is symmetric in
**x**and**y**:**x**·**y**=**y**·**x**. - It is
linear in its first argument: (*a***x**_{1}+*b***x**_{2}) ·**y**=*a***x**_{1}·**y**+*b***x**_{2}·**y**for any scalars a, b, and vectors**x**_{1},**x**_{2}, and**y**. - It is
positive definite : for all vectors**x**,**x**·**x**≥ 0 , with equalityif and only if **x**= 0.

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) **x**||, and to the angle θ between two vectors **x** and **y** by means of the formula

consisting of vectors in ℝ^{3} is ^{[1]}

Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector **L** in the Euclidean space, in the sense that

This property expresses the *completeness* of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.

Hilbert spaces are often taken over the *z*| which is defined as the square root of the product of z with its

If *z* = *x* + *iy* is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:

The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w:

This is complex-valued. The real part of ⟨*z*,*w*⟩ gives the usual two-dimensional Euclidean

A second example is the space ℂ^{2} whose elements are pairs of complex numbers *z* = (*z*_{1}, *z*_{2}). Then the inner product of z with another such vector *w* = (*w*_{1},*w*_{2}) is given by

The real part of ⟨*z*,*w*⟩ is then the four-dimensional Euclidean dot product. This inner product is *Hermitian* symmetric, which means that the result of interchanging z and w is the complex conjugate:

A **Hilbert space** *H* is a ^{[2]}

To say that *H* is a complex **inner product space** means that *H* is a complex vector space on which there is an inner product ⟨*x*,*y*⟩ associating a complex number to each pair of elements *x*, *y* of *H* that satisfies the following properties:

- The inner product of a pair of elements is equal to the
complex conjugate of the inner product of the swapped elements:

- The inner product is
linear in its first^{[nb 1]}argument. For all complex numbers*a*and*b*,

- The inner product of an element with itself is
positive definite :

- where and
- where

It follows from properties 1 and 2 that a complex inner product is

A real inner product space is defined in the same way, except that *H* is a real vector space and the inner product takes real values. Such an inner product will be bilinear: that is, linear in each argument.

The

and the distance *d* between two points *x*, *y* in *H* is defined in terms of the norm by

That this function is a distance function means firstly that it is symmetric in *x* and *y*, secondly that the distance between *x* and itself is zero, and otherwise the distance between *x* and *y* must be positive, and lastly that the *xyz* cannot exceed the sum of the lengths of the other two legs:

This last property is ultimately a consequence of the more fundamental

with equality if and only if *x* and *y* are

With a distance function defined in this way, any inner product space is a **pre-Hilbert space**.^{[3]} Any pre-Hilbert space that is additionally also a

The **Completeness** of *H* is expressed using a form of the *H*: a pre-Hilbert space *H* is complete if every

then the series converges in *H*, in the sense that the partial sums converge to an element of *H*.

As a complete normed space, Hilbert spaces are by definition also

The *l*^{2} consists of all **z** = (*z*_{1},*z*_{2},...) of complex numbers such that the

*l*^{2} is defined by

with the latter series converging as a consequence of the Cauchy–Schwarz inequality.

Completeness of the space holds provided that whenever a series of elements from *l*^{2} converges absolutely (in norm), then it converges to an element of *l*^{2}. The proof is basic in ^{[4]}

Other Languages

Afrikaans: Hilbert-ruimte

العربية: فضاء هيلبرت

asturianu: Espaciu de Hilbert

azərbaycanca: Hilbert fəzası

বাংলা: হিলবার্ট জগৎ

български: Хилбертово пространство

català: Espai de Hilbert

čeština: Hilbertův prostor

dansk: Hilbertrum

Deutsch: Hilbertraum

eesti: Hilberti ruum

Ελληνικά: Χώρος Χίλμπερτ

español: Espacio de Hilbert

Esperanto: Hilberta spaco

فارسی: فضای هیلبرت

français: Espace de Hilbert

galego: Espazo de Hilbert

한국어: 힐베르트 공간

հայերեն: Հիլբերտյան տարածություն

italiano: Spazio di Hilbert

עברית: מרחב הילברט

Кыргызча: Гильберт мейкиндиги

lietuvių: Hilberto erdvė

magyar: Hilbert-tér

Nederlands: Hilbertruimte

日本語: ヒルベルト空間

norsk: Hilbertrom

norsk nynorsk: Hilbertrom

oʻzbekcha/ўзбекча: Gilbert fazosi

ਪੰਜਾਬੀ: ਹਿਲਬਰਟ ਸਪੇਸ

پنجابی: ہلبرٹ سپیس

polski: Przestrzeń Hilberta

português: Espaço de Hilbert

română: Spațiu Hilbert

русский: Гильбертово пространство

Scots: Hilbert space

shqip: Hapësira e Hilbertit

Simple English: Hilbert space

slovenčina: Hilbertov priestor

slovenščina: Hilbertov prostor

کوردی: بۆشاییی ھیلبێرت

српски / srpski: Хилбертов простор

srpskohrvatski / српскохрватски: Hilbertov prostor

suomi: Hilbertin avaruus

svenska: Hilbertrum

Tagalog: Espasyong Hilbert

Türkçe: Hilbert uzayı

українська: Гільбертів простір

Tiếng Việt: Không gian Hilbert

粵語: 囂拔空間

中文: 希尔伯特空间