# Questions tagged [lp-spaces]

For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.

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### Rudin functional analysis: theorem 13.25 (involving functional calculus for unbounded operators)

Let $(\Omega, \mathfrak{R})$ be a measurable space, $H$ be a Hilbert space and let $E: \mathfrak{R}\to B(H)$ be a resolution of the identity. Let $f: \Omega \to \mathbb{C}$ be a measurable function ...
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### Convergence of integrals without the dominated convergence theorem

Let $t>0$ and $\{f_n\}_{n \geq 1}$ be a sequence of functions that do not converge pointwise to any integrable function, but where: \begin{equation} \int_0^t f_n(s) ds \rightarrow F(t) < \...
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### $L^{\infty}([a,b])$ and the Lebesgue-Stieltjes integral

Let $[a,b]$ be a compact interval in $\mathbb{R}$, and $L^{\infty}([a,b])$ the space of all lebesgue measurable functions on $[a,b]$ essentially bounded; my question is whether these functions are ...
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### How can you proove that every bounded function in $L^1[0;1]$ can be approximated by continuous function in $C[0;1]$?

Here my question, is this true that: Every bounded function in $L^1[0;1]$ can be approximated by continuous functions in $C[0;1]$ It seems to me true as we know that $C[0;1]$ is dense in $L^1[0;1]$, ...
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1 vote
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### Exercise 4.19 (1) of Brezis

I am trying to solve the following exercise of Brezis' book on Functional Analysis. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence in $L^p(\Omega)$ with $1 < p < \infty$ and let $f \in L^p(\Omega)$...
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### Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
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### problem on $L^2$ (pointwise) convergence and Carleson's Theorem

I am having some problem on my arguments and I want to see where it fails. It deals with pointwise convergence in $L^2[-\pi,\pi]$: pointwise convergence is given by Carleson's Theorem (so it is a hard ...
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### Is convolution integrable in $L_1(\mathbb{R}^n)$?

Let $|h(y)|\in L_1(\mathbb{R}^n)$, i.e. $\int\limits_{\mathbb{R}^n}|h(y)|\,dy<+\infty$. Consider the function $F(x)=\int\limits_{\mathbb{R}^n}|h(x-y)|\,dy$. It is known that $F(x)$ be bounded and ...
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### Intersection of compact sets in different spaces

Let $A$ be a compact set in $L^1$ and $B$ a compact set in $L^2$. Determine if $A \cap B$ is compact in $L^1$ or $L^2$ or both. My idea: Since $L^2 \subset L^1$ and $\Vert \cdot \Vert_2$ is stronger ...
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### Function goes to 0 if all derivatives are $L^2$

Suppose that all derivatives of some function $f$ are $L^2$, so that $\left\lVert\frac{\partial^kf}{\partial x^k}\right\rVert<\infty$ for all $k\ge0$. Then is it true that $f\to0$ as $x\to\infty$? ...
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### The space $\operatorname{Lip}_{b}(X)$ is dense in $L_1(X, \mu)$ w.r.t. $\|\cdot\|_{L_1}$
I'm trying to prove this well-known property. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space, $\operatorname{Lip}_{b}(X)$ the space of Lipschitz continuous bounded real-valued ...