1
vote
1answer
53 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
1
vote
1answer
23 views

How to prove $f(x)=e^{\frac{1}{x}}$ is continous in $(0,a), a>0 $ and $\int_{0}^{a}e^{\frac{y}{x}}dx, y>0$ does not exist

I would aprecciate any advice. I'm trying to prove that in the context of a measure space, $(X,B,\lambda)$ , with $X=(0, + \infty) $, $B$ the Borel sigma-algebra and $\lambda$ the Lebesgue measure, ...
1
vote
0answers
57 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
1
vote
1answer
20 views

$f \mapsto \int f^2$ is $L^1$-weakly lower semicontinuous

If $f \in C(\mathbb [0,1], \mathbb R)$ is $$ f \mapsto \int_0^1 f(t)^2\ dt$$ $L^1$-weakly lower semicontinuous? I.e. if $$\int_0^1 f_n g \rightarrow \int_0^1 f g$$ for every $g \in L^{\infty}$, then ...
2
votes
1answer
44 views

Integral of $L^2$ function is continuous

For $f\in L^2(\mathbb{R})$, denote $$s_N(x)=\dfrac{1}{2\pi}\int_{-N}^N\hat{f}(t)e^{ixt}dt.$$ I'd like to prove that the integral converges, and that $s_N$ is continuous. Since $f\in ...
2
votes
1answer
136 views

Continuous function bounded in $L^\infty$

Is a continuous (real-valued) function in $L^\infty$ a (everywhere-)bounded function?
1
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1answer
913 views

Using Lusin's Theorem to show that continuous functions are dense in $L^p$

Lusin's theorem says that in a finite measure space, given a measurable function $\varphi$, for every $\varepsilon \gt 0$ there exists a continuous function $g$ such that $$ \mu\left(\{x : ...