Which of These Definitions Is Correct for a Proximal Operator?

Question

For given function $g(W)$, where $W \in R^{M \times T}$. I have seen two different definition of proximal operator of it, but I don't know which one is correct ? One with $L_2$-norm, and second one with Frobenius norm

$$prox_g(W)={arg\,min}_u(g(u) + \frac{1}{2} |||u-W||_2^2 )$$

$$prox_g(W)={arg\,min}_u(g(u) + \frac{1}{2} |||u-W||_F^2 )$$

Answer 1

It depends on the Hilbert space structure you put on the domain of $g$.

Put differently: You may choose different scalar products for the space on which $g$ is defined and each gives rise to a valid definition of a proximal operator.