L^p^ → L^∞^

Proposition

Let

X

be a measure space, with positive measure

μ

. Assume

f \in L^{r} (μ)

for some real

r

, and

∥ f ∥_{\infty} < \infty

. We then assert that

∥ f ∥_{p} \to ∥ f ∥_{\infty}

p \to \infty

∥ f ∥_{\infty} = 0

, then

f = 0

a.e., and consequently,

∥ f ∥_{p} = 0

for all real positive

p

. The result then follows trivially.

Otherwise, put

F = \frac{f}{∥ f ∥_{\infty}}

, and let

0 < ε < 1

be given. Choose

0 < δ < ε

. Since

∥ F ∥_{\infty} = 1

, it follows that for sufficiently large

p \in R^{+}

{(\frac{1 - ε}{1 - δ})}^{p} < μ (| F |^{- 1} ((1 - δ, \infty))),

so that

1 - ε < {∥ (1 - δ) χ_{| F |^{- 1} ((1 - δ, \infty))} ∥}_{p} \leq {∥ F ∥}_{p} .

Next: since

| F | \leq 1

a.e., then

p > r

implies

| F |^{p} \leq | F |^{r}

a.e. and hence

∥ F ∥_{p} \leq ∥ F ∥_{r}^{r / p}

0 < ∥ F ∥_{r}^{r} < \infty

, it follows that

∥ F ∥_{r}^{r / p} \to 1

p \to \infty

. Thus,

∥ F ∥_{p} \to 1

p \to \infty

. It is then trivial to deduce

∥ f ∥_{p} \to ∥ f ∥_{\infty}

p \to \infty

/ / / /

Additionally, this relation still holds when

∥ f ∥_{\infty} = \infty

, but this verification is easier, and is thus left as an exercise for the reader.

Rudin, Walter. Real and Complex Analysis. 3rd ed, McGraw-Hill, 1987.