# Banach 23

The lecture: banach23

### Excerpts

Definitions

1. External direct sums of Hilbert spaces
2. Internal direct sums of Hilbert subspaces
3. Unitary operator between Hilbert spaces

Results

1. $T \in \mathscr{L}(H)$ implies $||T||^2 = ||T*T||$
2. $\mathscr{L}(H)$ is $C^*$ algebra with taking adjoint as involution.
3. $ = 0$ for all $x$ implies $T = 0$. (in the pf its supposed to say $ = etc.$, and are using sequilinear innerproduct $' := $.
4. $P \in \mathscr{L}(H)$ idempotent is orthogonal projection iff $P = P^*$
5. $T \in \mathscr{L}(H)$ is self adjoint iff $ \in \mathbb{R}$ $\forall x$.
6. $U \in \mathscr{L}(H_1, H_2)$ is unitary iff it is invertible and isometric.
7. $T \in \mathscr{L}(H)$ is normal iff $||Tx|| = ||T^*x||$ $\forall x$.