How to compare the nuclear seminorm of a matrix with the nuclear norm of the same matrix?

Question

I know that the nuclear norm $\| \cdot \|_*$ is defined as the sum of the singular values ($\sigma_i$) of the matrix, that is for an $n\times n$ matrix $L$, the nuclear norm is defined by

$$\| L \|_* = \sum_{i=1}^n \sigma_i(L),$$

and I read in a paper that this $\| L \|_* - <W,L>$ where $\| W \| \leq 1$ defines a semi-norm in $\mathbb{R}^{n \times n}$. My question is that how to compare the nuclear norm of $L$ and the nuclear semi-norm of $L$. I mean which one the greater than the other one, and how can I prove it?

Thanks in advance.

Answer 1

As you could see from this post, we have $$ \langle W, L \rangle \leq \|L\|_* \cdot \|W\| \leq \|L\|_* \cdot 1. $$