Suppose $X,Y$ are reflective Banach spaces. Let $Z=X\times Y$. The norm is defined as $$\|z\|=(\|x\|^p+\|y\|^p)^{\frac{1}{p}},\forall z=(x,y) \in X\times Y$$ here, $1\leq p\leq \infty$. Prove that $Z$ is reflective.
This is my functional analysis homework. And my idea is below:
$\forall F \in X^{**},\exists x\in X,s.t.F(f)=f(x), \forall f \in X^*$
$\forall G \in Y^{**},\exists y\in Y,s.t.G(g)=g(y), \forall g \in Y^*$
Now I want to prove $\forall H\in(X\times Y)^{**}, \exists z\in (X\times Y),s.t.H(h)=h(z),\forall h \in (X\times Y)^{*}$ (Another idea is to prove $\phi(X)=X^{**}$ directly, where $\phi$ is the canonical mapping)
And my problem is:How to use the condition that $X,Y$ are reflective Banach spaces? what's the relationship between $(X,Y)^{**}$ and $X^{**},Y^{**}$?
Thanks in advance:)