Unformatted text preview: K , then p ( x 1 + x 2 ) ≤ p ( x 1 ) + p ( x 2 ). 7. Let M be a closed subspace of a normed vector space X , let p ( x ) be a sublinear functional deﬁned on all of X , and let f ( m ) be a linear functional deﬁned on M that satisﬁes f ( m ) ≤ p ( m ) for all m ∈ M . Let y be a vector in x that is not in M . Prove that it is possible to deﬁne a linear functional g on the subspace [ M + y ] such that g ( x ) ≤ p ( x ) for all x ∈ [ M + y ] . 8. State the geometric version of the HahnBanach theorem, draw a diagram to illustrate it for the special case of a zerodimensional linear variety in R 2 , and prove it from the extension form of the theorem. You may take it as proved that the Minkowski functional of a convex set is sublinear. 1...
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 Fall '09
 Math, Vector Space, Hilbert space, normed vector space, Topological vector space, Banach space lp

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