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Difference between revisions of "Hahn-Banach theorem"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461301.png" /> be a linear manifold in a real or complex vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461302.png" />. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461303.png" /> is a [[Semi-norm|semi-norm]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461304.png" /> and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461305.png" /> is a [[Linear functional|linear functional]] defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461306.png" /> which satisfies
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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/h/h046/h046130/h0461307.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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Let $L$ be a linear manifold in a real or complex vector space $X$. Suppose $p$ is a [[semi-norm]] on $X$ and suppose $f$ is a [[linear functional]] defined on $L$ which satisfies
 +
\begin{equation}\label{eq:1}
 +
|f(x)| \le p(x)
 +
\end{equation}
 +
for every $x \in L$. Then $f$ can be extended to a linear functional $F$ on all of $X$ such that
 +
$$
 +
|F(x)| \le p(x)
 +
$$
 +
for all $x \in X$. Such is an extension is not uniquely determined.
  
for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461308.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h0461309.png" /> can be extended to a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613010.png" /> on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613011.png" /> such that
+
In the case of a real space $X$ the semi-norm can be replaced by a positively-homogeneous [[subadditive function]], and the inequality \ref{eq:1} by the one-sided inequality $f(x) \le p(x)$, which remains valid for the extended functional. If $X$ is a Banach space, then for $p(x)$ one can take $\Vert f \Vert_L \cdot \Vert x \Vert$, and then $\Vert F \Vert_X = \Vert f \Vert_L$. The theorem was proved by H. Hahn (1927), and independently by S. Banach (1929).
 
 
<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/h/h046/h046130/h04613012.png" /></td> </tr></table>
 
 
 
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613013.png" />. Such is an extension is not uniquely determined.
 
 
 
In the case of a real space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613014.png" /> the semi-norm can be replaced by a positively-homogeneous subadditive function, and the inequality (*) by the one-sided inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613015.png" />, which remains valid for the extended functional. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613016.png" /> is a Banach space, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613017.png" /> one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613018.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613019.png" />. The theorem was proved by H. Hahn (1927), and independently by S. Banach (1929).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hahn,  "Ueber lineare Gleichungsysteme in linearen Räume"  ''J. Reine Angew. Math.'' , '''157'''  (1927)  pp. 214–229</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  S. Banach,  "Sur les fonctionelles linéaires"  ''Studia Math.'' , '''1'''  (1929)  pp. 211–216</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  S. Banach,  "Sur les fonctionelles linéaires II"  ''Studia Math.'' , '''1'''  (1929)  pp. 223–239</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hahn,  "Ueber lineare Gleichungsysteme in linearen Räume"  ''J. Reine Angew. Math.'' , '''157'''  (1927)  pp. 214–229</TD></TR>
 +
<TR><TD valign="top">[2a]</TD> <TD valign="top">  S. Banach,  "Sur les fonctionelles linéaires"  ''Studia Math.'' , '''1'''  (1929)  pp. 211–216</TD></TR>
 +
<TR><TD valign="top">[2b]</TD> <TD valign="top">  S. Banach,  "Sur les fonctionelles linéaires II"  ''Studia Math.'' , '''1'''  (1929)  pp. 223–239</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613020.png" /> is called subadditive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613022.png" /> in its domain such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046130/h04613023.png" /> lies in its domain.
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A real-valued function $f$ is called subadditive if $f(x+y) \le f(x) + f(y)$ for all $x,y$ in its domain such that $x+y$ lies in its domain.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR>
 +
</table>

Latest revision as of 21:57, 20 December 2014


Let $L$ be a linear manifold in a real or complex vector space $X$. Suppose $p$ is a semi-norm on $X$ and suppose $f$ is a linear functional defined on $L$ which satisfies \begin{equation}\label{eq:1} |f(x)| \le p(x) \end{equation} for every $x \in L$. Then $f$ can be extended to a linear functional $F$ on all of $X$ such that $$ |F(x)| \le p(x) $$ for all $x \in X$. Such is an extension is not uniquely determined.

In the case of a real space $X$ the semi-norm can be replaced by a positively-homogeneous subadditive function, and the inequality \ref{eq:1} by the one-sided inequality $f(x) \le p(x)$, which remains valid for the extended functional. If $X$ is a Banach space, then for $p(x)$ one can take $\Vert f \Vert_L \cdot \Vert x \Vert$, and then $\Vert F \Vert_X = \Vert f \Vert_L$. The theorem was proved by H. Hahn (1927), and independently by S. Banach (1929).

References

[1] H. Hahn, "Ueber lineare Gleichungsysteme in linearen Räume" J. Reine Angew. Math. , 157 (1927) pp. 214–229
[2a] S. Banach, "Sur les fonctionelles linéaires" Studia Math. , 1 (1929) pp. 211–216
[2b] S. Banach, "Sur les fonctionelles linéaires II" Studia Math. , 1 (1929) pp. 223–239
[3] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[4] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)


Comments

A real-valued function $f$ is called subadditive if $f(x+y) \le f(x) + f(y)$ for all $x,y$ in its domain such that $x+y$ lies in its domain.

References

[a1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
How to Cite This Entry:
Hahn-Banach theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hahn-Banach_theorem&oldid=22537
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article