Let $E$ be a measurable set of finite measure and $1\leq p_1<p_2<\infty$. Consider the linear space $L^{p_2}(E)$ normed by $\|.\|_{p_1}$. Is this normed linear space a Banach space?
My professor hinted that the answer was no. So I developed the following example, taking $p_1=1$ and $p_2=2$: let $E=(0,1)$. Let the cauchy sequence of $L^{2}$ functions with respect to the $\|.\|_{1}$ norm be $f_n(x)=0$ for $x\in(0,\frac{1}{n})$, and $f_n(x)=\frac{1}{\sqrt{x}}$ for $x\in(\frac{1}{n},1)$.
With respect to the $\|.\|_1$ norm, it converges to $\frac{1}{\sqrt{x}}$, which is not an $L^2$ function over $(0,1)$. However, how do I show that there is no $L^2$ function such that $\{f_n\}$ converges to it with respect to the $\|.\|_1$ norm?