I have a problem with my assignment of Linear Analysis. It should be rather easy and straight-forward, but I have problems =(.
Let E and F be normed spaces. For $p \in [1,\infty]$, define the p-direct sum E$\oplus _p$F of E and F as the direct sum E$\oplus$F euipped with the norm $\left\lVert.\right\rVert _p$, where, for $(x,y) \in$ E$\oplus$F, we have $\left\lVert.\right\rVert _p$ = $\sqrt[p]{ \left\lVert x \right\rVert^p + \left\lVert y \right\rVert^p }$. if $p\in [1,\infty)$ and $\left\lVert.\right\rVert _\infty = max({\left\lVert x \right\rVert,\left\lVert y \right\rVert })$ Characterize (E$\oplus _p$F)* in terms of E* and F*.
I know it should be (E$\oplus _p$F)*= E * $\oplus _q$F*. I used this The dual of the direct sum, but now I do not know how to prove that it's an isometry too, and why it's q here, how to get that this exact norm. I realize that it should be like this but I will be glad to get some hint how to prove it. Any ideas. Thanks!