I found the following definition in this answer $$ \partial f(x) := \{x^* \in X^* \mid f(x') \ge f(x) + \langle x^*, x'-x\rangle\;\forall x' \in X\} $$
Can I define this $$ \langle A, B\rangle\ = Re(tr(B^H A))$$ and rewrite this definition
$\partial\|A\| := \{S\in \mathbb R^{m \times n} \quad \big| \quad \|B\|\geq\|A\| + tr((B-A)^T S),\forall B \in \mathbb R^{m \times n}\}$
in Characterization of the Subdifferential of Some Matrix Norms by G. A. Watson to
$\partial\|A\| := \{S\in \mathbb C^{m \times n} \quad \big| \quad \|B\|\geq\|A\| + Re(tr((B-A)^H S)),\forall B \in \mathbb C^{m \times n}\}$
I tried to find any paper where this was written in this way but found nothing. Any idea whether what I did here is correct?
I used this to find an element of $\partial\|A\|_2$ of a rank 1 hermitian matrix.