If $1\leq p < q <\infty$ and $E$ a subset of $\mathbb{R}$ with finite measure if we consider the space $L^q(E)$ is it a Banach space with the norm $||.||_p$. I know that $L^p$ space is a Banach space with respect to the $p$ norm.
And I know using Holder inequality that $||f||_p \leq ||f||_q (m(E))^{\frac{q-p}{pq}}$ so $E$ has finite measure then $L^q(E) \subset L^p(E)$. I was trying to construct a sequence $f_n \in L^q$ which is Cauchy with respect to the norm $||.||_p$ which does not converge but I could not? Or is that space a Banach space with smaller norm. I found similar questions but I did not find any counter example the solvers talk about atoms and the open map theorem, I am searching for a Cauchy sequence which diverge. The questions
Best Answer
If it is Banach then the identity map from $L^{q}$ with $L^{p}$ norm to $L^{q}$ with $L^{q}$ would be continuous by Open Mapping Theorem. So there would be a constant $C$ such that $\|f\|q \leq C \|f\|p$. Take the example $f_n=n^{1/q}I_{(0,1/n)}$ to get a contradiction. Aliter: if $f\in L^{p}\setminus L^{q}$ then $(fI_{1/n <|f|<n)})$ is a Cauchy sequence which is not convergent.