# Tagged Questions

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### Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
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### Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
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### How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
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### Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
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### Is $C[a,b]$ a closed linear subspace of $L^{p}(a,b)$

I am not sure about the last step of my proof: $(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). ...
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### Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
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### For which $L^p[0,1]$, $1\le p < \infty$ does the function $n^\alpha*\chi_{[0,1/n]}$ converge weakly to 0?

My hypothesis is that this function converges to 0 weakly iff $p < 1/\alpha$, but I am not sure how to prove this. We are working in the space $[0,1]$ with the Borel sets and Lebesgue measure. I ...
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### density of spaces, Lebesgue

My question is the following: Is the space $C^\infty(\bar{G})$ dense in $L^\infty(G)$ ? Assume that $G \subset \mathbb{R}^n$. I know this result holds, if we have $G$ instead of $\bar{G}$. But is it ...
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### Norm of Fredholm integral operator equals norm of its kernel?

Let $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt$, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$. Then it was fairly easy to see that $||T_k|| \le ||k||_{L^2}$, but now I was wondering how to show that ...
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### approximating a measurable function using simple functions

I would like to understand the proof that shows that for any measurable function, we can approximate it with a sequence of simple functions. The proof is presented in (among others): ...
Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
I believe I have a fundamental misunderstanding of the concept of the Vitali Covering Lemma. Definition - A closed bounded interval $[c, d]$ is said to be nondegenerate provided $c < d$. ...