Let $T: U \to V$ be a bounded linear operator between two real Hilbert spaces $U, V$. Then, is it true that $$\|Tu\| = \|(T^*T)^{1/2}u\| \ \text{for any} \ u \in U$$ where $T^*$ denotes the adjoint of $T$.
Now, my part is $$\|Tu\|^2 = \langle Tu, Tu\rangle = \langle T^*Tu, u\rangle = \langle (T^*T)^{1/2}u, (T^*T)^{1/2} u\rangle $$ $$= \| (T^*T)^{1/2}u\|^2 $$
Is my proof correct?
Yes your proof is correct. You should mention that $T^*T$ and $(T^*T)^{1/2}$ are self-adjoint.