Normed Vector Space

Related BrainMass Solutions

• Showing that convergence is not uniform

  Thus the sequence does not converge in the normed vector space (C ([0, 1]) , II·II infinity). b. Does the sequence converge to 0 in the normed vector space (C ([0, 1]) , II·II 2 (subscript 2) )?.

• One Dimensional Normed Linear Space : Completeness and Continuity

  Solution: a) E = normed one-dimensional vector space Let X = vector in E, that means X can be expressed as X = a*U (1) where U = one-dimensional basis of E, (a) belongs to the field on which E is defined as vector

• Convergent subsequences

  365389 Subsets of a normed vector space 1. If A and U are two subsets of a normed vector space, and U is open, show that A+U is open. Here A+U={a+u | a belongs to A and u belongs to U}. 2.

• Unit ball

  This is a proof regarding the unit ball in a normed vector space.

• Finite dimensional linear submanifold of N is complete.

  Since the normed linear space is complete, then any fundamental sequence has a vector whice is the limit of the sequence.

• Normed Linear Space : Hahn-Banach Theorem

  Suppose that E is a normed linear space. Prove that if E* is separable, then E is separable. Hint: Let be a countable dense subset of E*, and for each , let be a unit vector in E such that . Let M .

• Hahn Banach Theorem Application

  Since is a Banach space (normed vector space), the extension form of Hahn-Banach Theorem tells us that if for any , then there are extended linear funcationals : K such that for all and for all .

• g is a linear combination of f1, f2,...,fn iff g(N) = {0}

  Space Suppose that Ɛ is a vector space, nЄN, and f1, f2,...

• Topological Vector Space : Closed Kernel

  Prove that a linear functional f on T is continuous if and only if ker(f) is closed. let f: X ---> R be a non-zero functional from X, a normed space (any topological vector space will do).