Question

Prove the following result:

 Prove the following result:
Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set equation of a non-zero real numbers with equation and an orthonormal set equation in H such that


equation
Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

09 Jan 2026
Answer :
Word Count : 460
Let (A) be a non-zero compact self-adjoint operator on a Hilbert space (H) over (\mathbb{K}). We want to show the existence of a finite set of non-zero real numbers ({r_1, r_2, \dots, r_n}) with (|r_1| \ge |r_2| \ge \dots \ge |r_n|) and an orthonormal set ({w_1, w_2, \dots, w_n}) in (H) such that [ A(x) = \sum_{i=1}^n r_i \langle x, w_i\rangle w_i, \quad x \in H. ] --- Step 1: Spectral theorem for compact self-adjoint operators Since (A) is self-adjoint and compact, by the spectral theorem for compact self-adjoint operators, the spectrum (\sigma(A)) consists of a sequence of real eigenvalues (possibly accumulating at 0) and 0 itself. That is, all non-zero elements of (\sigma(A)) are eigenvalues of (A), and there exists an orthonormal set of eigenvectors corresponding to these eigenvalues. *Here, the fact that (A) is self-adjoint ensures that eigenvalues are real, and
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