Answer to Question #272582 in Functional Analysis for Prathibha Rose

Question

Show that the self adjoint operator is continuous map

Answer 1

A linear map T : X → Y is continuous if and only if its operator norm is finite

To prove that T^∗ is continuous, prove that it is bounded. From Cauchy-Schwarz-Bunyakowsky

$|T^*y|^2 = |\langle T^* y, T ^∗ y\rangle_X| = |\langle y, TT ^∗ y\rangle_Y| ≤ |y| · |T T^ ∗ y| ≤ |y| · |T| · |T ^∗ y|$

where |T| is the operator norm. For $T ^∗y \neq 0$ , divide by it to find

$|T^ ∗ y| ≤ |y| · |T|$

Thus,$|T^ ∗ | ≤ |T|$ . In particular, $T^ ∗$ is bounded.