# Frobenius Norm with Unitary Operators

For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this pretty easily if an inner product space exists. For example, $\|A\| = \sqrt{\langle A,A\rangle}$ and $\|UA\| = \sqrt{\langle UA,UA\rangle} = \sqrt{\langle A,U^*UA\rangle} = \sqrt{\langle A,A\rangle} = \|A\|$. However, I'm not sure if I evoke the inner product space if I'm just told that the Frobenius norm exists. Is this the appropriate approach?