$L^p$ continuous mapping problem

Question

Let $f ∈ L_1 ∩ L_4$ (on some measure space). Prove that the function $[1,4] → R$ given by $p → ∥f∥_p$ is continuous. This is a qualifying exam problem and I am not sure what to use. All I can think of is Holder and that doesn't seem like much help. I would appreciate any help.

Thanks

In the title you use a superscript for Lp, but in the question you use subscripts for L1 and L4, mind clarifying? — Squirtle, May 16 '14 at 00:55

score 1 · Accepted Answer · answered May 16 '14 at 03:54

It is clearly true if $f$ is a simple function. For general $f \in L^1 \cap L^4$, find a sequence of simple functions $f_n$ such that $f_n \to f$ in $L^1$ and $f_n \to f$ in $L^4$. Show that $\|f_n\|_p \to \|f\|_p$ uniformly in $p \in [1,4]$. For the last part, use Holder's inequality to show that $\|f-f_n\|_p \le \|f-f_n\|_1 ^{1-\theta} \|f-f_n\|_4^\theta$ for $\theta$ satisfying $\frac1p = \frac{1-\theta}1 + \frac\theta4$.

$L^p$ continuous mapping problem

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