For a unitarily invariant norm on $\mathbb C^n$, how do I show that $||x||=||x||_2||e_1||$? I can show that $||e_1||=1$ for the Euclidean norm by definition, and is therefore unitarily invariant, does that suffice as proof?
I've been told that the only unitarily invariant norm on $\mathbb C^n$ satisfying the above equality (that $||e_1||=1$) is the Euclidean norm, would someone help me understand why that is?
Note that $e_1$ is the elementary $n\times1$ col vector, $[1,0,0,...0]^T$