Norm of a matrix and its conjugate.

Question

If $A$ is a $m \times n$ matrix and $p, q \geq 1$ are conjugates satisfying $\frac{1}{p}+\frac{1}{q} = 1$, then prove that $$\|A\|_p=\|A^t\|_q$$ where $A^t$ is the transpose of $A$.

Any hint or please provide the solution?

Answer 1

Hint 1: Let $x$ be a vector. Then $\|x\|_p = \sup_{\|y\|_q \le 1} y^Tx$.

Hint 2: $\|A\|_p = \sup_{\|x\|_p\le 1} \|Ax\|_p$.

Hint 3: for sets $C,D$ and function $f:C\times D\to \mathbb R$ it holds $$\sup_{c\in C}\sup_{d\in D} f(c,d) = \sup_{d\in D}\sup_{c\in C} f(c,d)$$