Hahn-Banach theorem

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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 $$\label{eq:1} |f(x)| \le p(x)$$ 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)

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.