Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Question

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$!

Unfortunately I haven't many ideas for this task. I know that the definition of the operatornorm for a matrix is: $$\|A\|=\sup\{\|Ax\| \,\big{|}\, \|x\|=1\}$$ but this is not useful for me. Any hints or explanations for this?

Answer 1

Hint: For any $x$ with $\|x\|_1=1$, we have $\|x\|_\infty \leq 1$.