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# Hahn Banach Theorem Application

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Suppose that is a Banach space over K. A subspace M of is said to be complemented in if there exists a subspace N of such that =M N, that is if , then there exists in M and in N such that , and M N . Prove that each finite dimensional subspace of is complemented in .

Hint: Suppose that M is a finite dimensional subspace of and is a basis for M. Define linear functionals : M K by for in K, k=1,2,...,n. Apply the Hahn-Banach Theorem, and look at the kernels of the linear functionals.

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#### Solution Preview

Suppose that is a Banach space over K. A subspace M of is said to be complemented in if there exists a subspace N of such that =M N, that is if , then there exists in M and in N such that , and M N . Prove that each finite dimensional subspace of is complemented in .

Hint: Suppose that M is a finite dimensional subspace of and is a basis for M. Define linear functionals : M K by for ...

#### Solution Summary

A Hahn Banach Theorem application problem is solved.

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