Does the matrix norm inequality or the Cauchy-Schwarz inequality hold for L2,1 norms

Question

I read here https://statweb.stanford.edu/~souravc/Lecture32.pdf that Cauchy-Schwarz inequality holds for the Hilbert-Schmit or Frobenius norms. I wanted to know if the same holds for other norms too specifically the L2,1 norm.

Answer 1

Since both Hilbert-Schmidt norm and $L^2$-norm arise from an inner product (more generally, semi-inner product), Cauchy-Schwarz holds in those cases. But if we consider $L^p([0,1])$, $p\neq 2$ for example, there does not exist an inner product to apply Cauchy-Schwarz inequality, because $L^p$-norm does not satisfy the parallelogram law $$ \|x+y\|_p^2+\|x-y\|_p^2 = 2(\|x\|_p^2+\|y\|_p^2), $$which should be true for all norms obtained from an inner product.