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A finite non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842501.png" /> on a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842502.png" /> (over the field of real or complex numbers) satisfying the following conditions:
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A finite non-negative function $p$ on a [[Vector space|vector space]] $E$ (over the field of real or complex numbers) satisfying the following conditions:
 
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\begin{equation}
<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/s/s084/s084250/s0842503.png" /></td> </tr></table>
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p(\lambda x) = |\lambda|p(x),\quad p(x+y)\leq p(x) + p(y)
 
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\end{equation}
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842504.png" /> and all scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842505.png" />. An example of a semi-norm is a [[Norm|norm]]; the difference is that a semi-norm may have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842506.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842507.png" />. If a semi-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842508.png" /> is defined on a vector space and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842509.png" /> is a [[Linear functional|linear functional]] on a subspace obeying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425010.png" />, then this functional can be extended to the entire space so that the extension satisfies the same condition (the Hahn–Banach theorem). In mathematical analysis one frequently encounters separable topological vector spaces (cf. [[Topological vector space|Topological vector space]]) in which an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425011.png" />-neighbourhood basis exists whose elements are convex sets. Such spaces are said to be locally convex. An open convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425012.png" />-neighbourhood in a locally convex space is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425014.png" /> is a continuous semi-norm. Nevertheless, in mathematical analysis one also encounters topological vector spaces (including spaces with a metrizable topology) on which there exist no non-trivial semi-norms. The simplest examples of such spaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425016.png" />.
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for all $x,y\in E$ and all scalars $\lambda$. An example of a semi-norm is a [[Norm|norm]]; the difference is that a semi-norm may have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842506.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842507.png" />. If a semi-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842508.png" /> is defined on a vector space and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s0842509.png" /> is a [[Linear functional|linear functional]] on a subspace obeying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425010.png" />, then this functional can be extended to the entire space so that the extension satisfies the same condition (the Hahn–Banach theorem). In mathematical analysis one frequently encounters separable topological vector spaces (cf. [[Topological vector space|Topological vector space]]) in which an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425011.png" />-neighbourhood basis exists whose elements are convex sets. Such spaces are said to be locally convex. An open convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425012.png" />-neighbourhood in a locally convex space is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425014.png" /> is a continuous semi-norm. Nevertheless, in mathematical analysis one also encounters topological vector spaces (including spaces with a metrizable topology) on which there exist no non-trivial semi-norms. The simplest examples of such spaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084250/s08425016.png" />.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Topological vector spaces" , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Topological vector spaces" , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1979)</TD></TR></table>

Revision as of 14:32, 4 December 2012

A finite non-negative function $p$ on a vector space $E$ (over the field of real or complex numbers) satisfying the following conditions: \begin{equation} p(\lambda x) = |\lambda|p(x),\quad p(x+y)\leq p(x) + p(y) \end{equation} for all $x,y\in E$ and all scalars $\lambda$. An example of a semi-norm is a norm; the difference is that a semi-norm may have with . If a semi-norm is defined on a vector space and if is a linear functional on a subspace obeying the condition , then this functional can be extended to the entire space so that the extension satisfies the same condition (the Hahn–Banach theorem). In mathematical analysis one frequently encounters separable topological vector spaces (cf. Topological vector space) in which an -neighbourhood basis exists whose elements are convex sets. Such spaces are said to be locally convex. An open convex -neighbourhood in a locally convex space is of the form , where is a continuous semi-norm. Nevertheless, in mathematical analysis one also encounters topological vector spaces (including spaces with a metrizable topology) on which there exist no non-trivial semi-norms. The simplest examples of such spaces are , .

References

[1] N. Bourbaki, "Topological vector spaces" , Springer (1987) (Translated from French)
[2] W. Rudin, "Functional analysis" , McGraw-Hill (1979)
How to Cite This Entry:
Semi-norm. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Semi-norm&oldid=29075
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article