Namespaces
Variants
Actions

Difference between revisions of "Pre-Hilbert space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
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:
+
<!--
 +
p0742901.png
 +
$#A+1 = 18 n = 0
 +
$#C+1 = 18 : ~/encyclopedia/old_files/data/P074/P.0704290 Pre\AAhHilbert space
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
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" />);
+
{{TEX|auto}}
 +
{{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" />;
+
A [[Vector space|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:
  
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" />.
+
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 $);
  
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]].
+
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|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.
+
A function $  ( x, y) $
 +
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
+
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
  
<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>
+
$$
 +
\| x+ y \|  ^ {2} + \| x \cdot y \|  = \
 +
2 ( \| x \|  ^ {2} + \| y \|  ^ {2} ) .
 +
$$
  
 
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=48275
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article