Spectrum of Self-Adjoint Bounded Linear Operator is Real and Closed/Proof 1

Theorem

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $T : \HH \to \HH$ be a bounded self-adjoint operator.

Let $\map \sigma T$ be the spectrum of $T$.

Then:

$\map \sigma T \subseteq \R$

Proof

This follows from:

Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed

Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator

$\blacksquare$