Problem 4 chapter 2: functional analysis (Rudin)

Question

$L^1$, $L^2$: usual Lebesgue spaces on the unit interval. Show that $L^2$ is of the first category (meager) in $L^1$, in three ways:

(a) Show that $F_n:=\{f:\int|f|^2 \leq n\}$ is closed in $L^1$ but has empty interior.

(b)Put $g_n=n$ on $[0,n^{-3}]$, and show that $\int fg_n \rightarrow 0$ $\forall f \in L^2$ , but not for every $f\in L^1$.

(c) Note that the inclusion map of $L^2$ into $L^1$ is continuous but not onto.

Do the same for $L^p$ and $L^q$ if $p<q$.

Answer 1

For each question, there are two sub-questions: 1. Why the assertion is true? and 2. Why does this one give the result?

a. 1. Show sequential closeness (as we are in a metric context) by Fatou's lemma. What happens if we assume that $F_n$ has a non-empty interior (in $L^1$)? 2 It's by definition.

b. 1. Use the fact that $\{f_n\}$ is bounded in $L^2$ and that simple functions are dense in $L^2$.

c. 1. It follows from a well-known integral inequality. Take an integrable function which is not square integrable. 2. There is a related theorem earlier in the book.