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

We are given an orthonormal set \( \{x_k\}_{k=1}^\infty \) in a Hilbert space \( X \), and \( Y \) is the linear span of the first \( n \) vectors. The map \( S_n(x) = \sum_{k=1}^n \langle x, x_k \rangle x_k \) represents the projection of \( x \) onto the subspace \( Y \).

Step 1: Orthogonal Projection onto \( Y \)
Since \( \{x_k\}_{k=1}^\infty \) is an orthonormal set, \( S_n(x) \) represents the orthogonal projection of \( x \) onto the subspace \( Y \). Therefore, (A) is TRUE.

Step 2: Orthogonal Projection onto \( Y^\perp \)
\( S_n(x) \) is not the projection onto \( Y^\perp \), it is the projection onto \( Y \), so (B) is FALSE.

Step 3: Orthogonality of \( x - S_n(x) \) and \( S_n(x) \)
By the properties of orthogonal projections, \( (x - S_n(x)) \) is orthogonal to \( S_n(x) \), so (C) is TRUE.

Step 4: The sum \( \sum_{k=1}^n \langle x, x_k \rangle^2 \)
The sum \( \sum_{k=1}^n \langle x, x_k \rangle^2 \) is not equal to \( \| x \|^2 \) for all \( x \in X \); it only gives the squared norm of the projection of \( x \) onto the span of \( \{ x_k \}_{k=1}^n \). Therefore, (D) is FALSE.

Thus, the correct answers are:

\[ \boxed{(A)} \quad S_n(x) \text{ is the orthogonal projection of } x \text{ onto } Y. \]
\[ \boxed{(C)} \quad (x - S_n(x)) \text{ is orthogonal to } S_n(x) \text{ for all } x \in X. \]