In the space of n*n matrices, for any m<=n can we define the sum of the largest m singular values for a matrix as a norm?

Question

The question is written in the title, I met this question when I tried to prove a little problem in my work. It is proved that both the largest singular value and the sum of all singular values can be norms of the n*n matrices space. So, I think this statement is also right. In fact, I only need the proof of the triangle inequality.