Let $E$ be a measurable set of finite measure and $1\leq a&lt;b&lt;\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space?  I think this is wrong, so I tried to find a counterexample but it is hard to find one...
$E$ is a measurable set of finite measure and $1 \le p_{1} &lt; p_{2} &lt; \infty$. Consider the linear space $L^{p_{2}}(E)$ normed by $||.||_{p_{1}}$. Is this space Banach?
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