(Solution) - Consider the subspace W u a 0 u b -(2025 Original AI-Free Solution)

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Consider the subspace W = {u(a) = 0 = u(b)} of the vector space C0[a, b] with the usual L2 inner product.
(a) Show that W has a complementary subspace of dimension 2.
(b) Prove that there does not exist an orthogonal complement to W. Thus, an infinite-dimensional subspace may not admit an orthogonal complement!