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Difference between revisions of "Convergence in norm"

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Convergence of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026070/c0260701.png" /> in a normed vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026070/c0260702.png" /> to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026070/c0260703.png" />, defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026070/c0260704.png" /> if
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{{MSC|46Bxx}}
 
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{{TEX|done}}
<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/c/c026/c026070/c0260705.png" /></td> </tr></table>
 
 
 
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026070/c0260706.png" /> is the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026070/c0260707.png" />.
 
  
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Convergence of a sequence $(x_n)$ in a normed vector space $X$ to an element $x$, defined in the following way: $x_n \rightarrow x$ if
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$$
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\text{$\left\| x_n - x \right\| \rightarrow 0$ as $n\rightarrow\infty$.}
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$$
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Here $\left\|\cdot\right\|$ is the norm in $X$.
  
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====Comments====
  
====Comments====
 
 
See also [[Convergence, types of|Convergence, types of]].
 
See also [[Convergence, types of|Convergence, types of]].

Latest revision as of 16:33, 4 May 2012

2010 Mathematics Subject Classification: Primary: 46Bxx [MSN][ZBL]

Convergence of a sequence $(x_n)$ in a normed vector space $X$ to an element $x$, defined in the following way: $x_n \rightarrow x$ if $$ \text{$\left\| x_n - x \right\| \rightarrow 0$ as $n\rightarrow\infty$.} $$ Here $\left\|\cdot\right\|$ is the norm in $X$.

Comments

See also Convergence, types of.

How to Cite This Entry:
Convergence in norm. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Convergence_in_norm&oldid=25986
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article