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A linear functional defined on a subspace of a vector space V and which is dominated by a sublinear function defined on V has a linear extension which is also dominated by the sublinear function.
References
Casti, J. L. "The Hahn-Banach Theorem." Ch. 4 in Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: Wiley, pp. 155-205, 2000.
Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.
Credit to Eric W. Weisstein. From MathWorld--A Wolfram Web Resource.
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