Proving that the operator $G: L_p[0,1] \to L_q[0,1]; G(x)=x$ is linear, well defined and continuous. ($p\geq q$)
Linearity is trivial. The well-defined part, with continuity and norm isn't. (for me)
I have to prove: $$\|G(x)\|_{L_q[a,b]}<+\infty ; x \in L_p[a,b]$$ or $$\int_{a}^{b}|x(t)|^qdt<\infty$$ having that $\int_{a}^{b}|x(t)|^pdt<\infty$. We have done Holder equality, but I dont think I can use that directly seeing as $1/p + 1/q = 1$ doesn't necessarily have to be true.
