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(1)Let X be a Banach space and Y ⊂ X a closed vector subspace

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Let X be a Banach space and Y ⊂ X a closed vector subspace. We call Y a complemented subspace if and only if there exists a closed vector subspace Z ⊂ X satisfying Y ∩ Z = {0} and X = Y + Z.

Part A. Prove that every closed subspace of a Hilbert space is a complemented subspace.

Part B. Let X be a Banach space and Y ⊂ X a closed complemented subspace. Take a closed vector subspace Z ⊂ X satisfying Y ∩ Z = {0} and X = Y + Z.

• Let Y ⊕ Z be the Banach space with k(y, z)k = kyk + kzk. Prove that the linear map θ : Y ⊕ Z → X : θ(y, z) = y + z has a bounded inverse.

• Deduce the existence of α > 0 such that ky + zk ≥ αkyk for all y ∈ Y , z ∈ Z.

Part C. Let X be a Banach space and Y ⊂ X a vector subspace. Prove that the following two statements are equivalent.

• Y is closed and complemented.

• There exists a bounded linear map E : X → X satisfying E ◦ E = E and Y = E(X).

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