Question

Let \( \{x_k\}_{k=1}^\infty \) be an orthonormal set of vectors in a real Hilbert space \( X \) with inner product \( \langle \cdot, \cdot \rangle \). Let \( n \in \mathbb{N} \), and let \( Y \) be the linear span of \( \{ x_k \}_{k=1}^n \) over \( \mathbb{R} \). For \( x \in X \), let \[ S_n(x) = \sum_{k=1}^n \langle x, x_k \rangle x_k. \] Then, which of the following is/are TRUE?

• A. \( S_n(x) \) is the orthogonal projection of \( x \) onto \( Y \)
• B. \( S_n(x) \) is the orthogonal projection of \( x \) onto \( Y^\perp \)
• C. \( (x - S_n(x)) \) is orthogonal to \( S_n(x) \) for all \( x \in X \)
• D. \( \sum_{k=1}^n \langle x, x_k \rangle^2 = \|x\|^2 \) for all \( x \in X \)

Hint:

The orthogonal projection of a vector onto a subspace is the sum of the components along the orthonormal basis of the subspace. Remember that \( S_n(x) \) is the projection onto the span of the first \( n \) vectors.

Answer 1

The Correct Option is A, C

To solve this problem, we need to analyze each of the given statements regarding the series \( S_n(x) \) defined as:

S_n(x) = \sum_{k=1}^n \langle x, x_k \rangle x_k.

Here, \( \{ x_k \}_{k=1}^n \) is an orthonormal set in a real Hilbert space \( X \), and \( Y \) is the linear span of these vectors.

1. Verify if \( S_n(x) \) is the orthogonal projection of \( x \) onto \( Y \):
   The orthogonal projection of a vector \( x \) onto a subspace \( Y \) is the vector in \( Y \) that is closest to \( x \). For an orthonormal basis \(\{x_k\}_{k=1}^n\), the projection is given by \sum_{k=1}^n \langle x, x_k \rangle x_k. Thus, \( S_n(x) \) is indeed the orthogonal projection of \( x \) onto \( Y \).
2. Check if \( S_n(x) \) is the orthogonal projection onto \( Y^\perp \):
   By definition, \( S_n(x) \) lies in \( Y \), not in the orthogonal complement \( Y^\perp \). Therefore, this statement is false.
3. Verify if \( (x - S_n(x)) \) is orthogonal to \( S_n(x) \):
   If \( S_n(x) \) is the orthogonal projection of \( x \) onto \( Y \), the vector \( x - S_n(x) \) must be orthogonal to every vector in \( Y \), including \( S_n(x) \). This confirms the statement is true.
4. Check the validity of \( \sum_{k=1}^n \langle x, x_k \rangle^2 = \|x\|^2 \) for all \( x \):
   This is generally true only if \( x \) lies in the span of the orthonormal set \(\{x_k\}_{k=1}^n\). In general, Parseval's identity states: \|x\|^2 = \sum_{k=1}^\infty \langle x, x_k \rangle^2. Because the identity holds for the entire space but not for any finite n, this statement is false.

Based on the analysis, the true statements are:

• \( S_n(x) \) is the orthogonal projection of \( x \) onto \( Y \).
• \( (x - S_n(x)) \) is orthogonal to \( S_n(x) \) for all \( x \in X \).