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A [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742901.png" /> over the field of complex or real numbers equipped with a scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742903.png" />, satisfying the following conditions:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742908.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p0742909.png" />);
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{{TEX|done}}
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429011.png" />;
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A [[Vector space|vector space]]  $  E $
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over the field of complex or real numbers equipped with a scalar product  $  E \times E \rightarrow \mathbf C $,
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$  x \times y \rightarrow ( x , y ) $,
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satisfying the following conditions:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429012.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429013.png" />.
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1)  $  ( x + y , z ) = ( x , z ) + ( y , z ) $,
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$  ( \lambda x , y ) = \lambda ( x , y ) $,
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$  ( y , x ) = \overline{ {( x , y ) }}\; $,
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$  x , y , z \in E $,
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$  \lambda \in \mathbf C $(
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$  \mathbf R $);
  
On a pre-Hilbert space a norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429014.png" /> is defined. The completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429015.png" /> with respect to this norm is a [[Hilbert space|Hilbert space]].
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2)  $  ( x , x ) \geq  0 $
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for  $  x \in E $;
  
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3)  $  ( x, x) = 0 $
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if and only if  $  x = 0 $.
  
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On a pre-Hilbert space a norm  $  \| x \| = ( x , x )  ^ {1/2} $
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is defined. The completion of  $  E $
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with respect to this norm is a [[Hilbert space|Hilbert space]].
  
 
====Comments====
 
====Comments====
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429016.png" /> as above is also called an [[Inner product|inner product]]. If it satisfies only 1) and 2) it is sometimes called a pre-inner product. Accordingly, pre-Hilbert spaces are sometimes called inner product spaces, while vector spaces with a pre-inner product are also called pre-inner product spaces.
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A function $  ( x, y) $
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as above is also called an [[Inner product|inner product]]. If it satisfies only 1) and 2) it is sometimes called a pre-inner product. Accordingly, pre-Hilbert spaces are sometimes called inner product spaces, while vector spaces with a pre-inner product are also called pre-inner product spaces.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429017.png" /> is a normed linear space, then it has an inner product generating the norm if (and only if) the norm satisfies the parallelogram law
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If $  ( E, \| \cdot \| ) $
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is a normed linear space, then it has an inner product generating the norm if (and only if) the norm satisfies the parallelogram law
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074290/p07429018.png" /></td> </tr></table>
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$$
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\| x+ y \|  ^ {2} + \| x \cdot y \|  = \
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2 ( \| x \|  ^ {2} + \| y \|  ^ {2} ) .
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$$
  
 
For the characterizations of inner product spaces see [[#References|[a1]]], Chapt. 4.
 
For the characterizations of inner product spaces see [[#References|[a1]]], Chapt. 4.

Latest revision as of 08:07, 6 June 2020


A vector space $ E $ over the field of complex or real numbers equipped with a scalar product $ E \times E \rightarrow \mathbf C $, $ x \times y \rightarrow ( x , y ) $, satisfying the following conditions:

1) $ ( x + y , z ) = ( x , z ) + ( y , z ) $, $ ( \lambda x , y ) = \lambda ( x , y ) $, $ ( y , x ) = \overline{ {( x , y ) }}\; $, $ x , y , z \in E $, $ \lambda \in \mathbf C $( $ \mathbf R $);

2) $ ( x , x ) \geq 0 $ for $ x \in E $;

3) $ ( x, x) = 0 $ if and only if $ x = 0 $.

On a pre-Hilbert space a norm $ \| x \| = ( x , x ) ^ {1/2} $ is defined. The completion of $ E $ with respect to this norm is a Hilbert space.

Comments

A function $ ( x, y) $ as above is also called an inner product. If it satisfies only 1) and 2) it is sometimes called a pre-inner product. Accordingly, pre-Hilbert spaces are sometimes called inner product spaces, while vector spaces with a pre-inner product are also called pre-inner product spaces.

If $ ( E, \| \cdot \| ) $ is a normed linear space, then it has an inner product generating the norm if (and only if) the norm satisfies the parallelogram law

$$ \| x+ y \| ^ {2} + \| x \cdot y \| = \ 2 ( \| x \| ^ {2} + \| y \| ^ {2} ) . $$

For the characterizations of inner product spaces see [a1], Chapt. 4.

References

[a1] V.I. Istrăţescu, "Inner product structures" , Reidel (1987)
[a2] W. Rudin, "Functional analysis" , McGraw-Hill (1979)
[a3] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Pre-Hilbert space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-Hilbert_space&oldid=15523
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article