An infinite orthonormal set {x_k} lies in a real Hilbert space X. Let Y=span{x₁,…,x_n} and $S_nx=\sum_{k=1}^n\langle x,x_k\rangle x_k$. Which statements hold?
Correct answer: (A) S_nx is the projection onto Y.; (C) x−S_nx is orthogonal to S_nx.
Explanation
S_nx lies in Y and subtracting it leaves zero inner product with each basis x_k, so it is the orthogonal projection onto Y and the remainder is orthogonal to it. A,C hold. Taking x=x_(n+1) gives left sum zero but ‖x‖²=1, disproving D; B identifies the wrong projection. **A,C**.
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