Trace class and Hilbert Schmidt Ideals (equalities)

We have begun looking at trace class and Hilbert Schmidt ideals in lectures today. In particular, looking at pages 206-212 of Reed and Simon book 1.

My lecture asserted the following two equivalences in class today and I don't understand the details.

If we let $\{ b_n \}$ be an orthonormal basis, it was asserted that the Hilbert space norm of $\left| A \right|$ was equal to the Hilbert space norm of $A$. That is, $$\left( \sum_{n=1}^{\infty} \| \left| A \right| b_n \|^2 \right)^{\frac{1}{2}} = \left( \sum_{n=1}^{\infty} \| A b_n \|^2 \right)^{\frac{1}{2}}.$$

And the second assertion was that the square of the Hilbert space norm of $\left| A \right|^{\frac{1}{2}}$ was the trace norm of $A$.

$$\|\,|A|\,x\|^2=\langle\,|A|x,|A|x\rangle=\langle |A|^2x,x\rangle=\langle A^*Ax,x\rangle=\langle Ax,Ax\rangle=\|Ax\|^2.$$ And $$\|\,|A|^{1/2}\,\|_2^2=\text{Tr}((|A|^{1/2})^*|A|^{1/2})=\text{Tr}(|A|).$$