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Hahn-Banach Theorem is one of the core theorems of linear functional analysis. The Hahn-Banach Theorem in Vector Space is also called Analytic Form of Hahn-Banach Theorem. Two corollaries are especially important: Hahn-Banach Theorem in Normed Vector Space, and Geometric Form of Hahn-Banach Theorem.
Analytic Form
定理.[Hahn-Banach Theorem in Real Vector Space]
Let XX be a real vector space, and pp is a sublinear functional in XX, that is, p:X→Rp:X→R is a function satisfies the following properties:
p(αx)=αp(x),∀α>0 and x∈X,p(x+y)⩽p(x)+p(y)∀x,y∈X.p(αx)=αp(x),∀α>0 and x∈X,p(x+y)⩽p(x)+p(y)∀x,y∈X.
And let YY be a subspace of XX, l:Y→Rl:Y→R is a linear functional in YY which satisfies
l(y)⩽q(y),∀y∈Y.l(y)⩽q(y),∀y∈Y.
Then there exists a linear functional ˜j:X→R˜j:X→R, such that
˜l(y)=l(y),∀y∈Y. and ˜l(y)⩽p(x),∀x∈X.˜l(y)=l(y),∀y∈Y. and ˜l(y)⩽p(x),∀x∈X.