Is $(L^q(E),\|.\|_p)$ a Banach space?

Question

Let $E$ be a measurable subset of $\mathbb R$ with finite measure and let $1\leq p<q<\infty$. We know that $L^q(E)\subset L^p(E)$. Can we say that $(L^q(E),\|.\|_p)$ a Banach space? I feel there is a Cauchy sequence in $(L^q(E),\|.\|_p)$ which does not converge to $(L^q(E),\|.\|_p)$. Any help will be appreciated.

Answer 1

Let $f(x)=x^{-a}$ for $0<x<1$ and $0$ elsewhere, where $\frac 1 q <a <\frac 1 p$. Then $fI_{\{|x| \leq n\}} I_{\{|f(x)| \leq n\}}$ is a Cauchy sequence which is not convergent. [It converges in p-norm to $f$ which is not in $L^{q}$].