Hahn banach theorem application

Question

Problem:

Hahn Banach Theorem Application

Suppose that ε is a Banach space over K. A subspace M of is said to be complemented in if there exists a subspace N of ε such that ε =M ⊕ N, that is if x ∈ ε , then there exists in M and in N such that x = y+z,and M∩N = {0}. Prove that each finite dimensional subspace of is complemented in ε .

Hint: Suppose that M is a finite dimensional subspace of ε and {x₁.x₂,....xn} is a basis for M. Define linear functionals ƒk :M→K by ƒk(λ₁x₁ + λ₂x₂ + ,....,+λ_nx_n) = λk for λ₁λ₂,....λ . in K, k=1,2,...,n. Apply the Hahn-Banach Theorem, and look at the kernels of the linear functionals.