# 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 class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a problem is a $L^p$ space.

3,274 questions
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### Examples that are not Lebesgue integrable for any $p$

I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but ...
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### Limit problem for $L^p$ function

I am having problems with proving the following: Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that $$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$$ ...
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### How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
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### Question from Folland, criteron for a function to belong to $L^p$

This question is from Folland 6.38, Show that $f \in L^p$ iff $\sum_{k=-\infty}^ {\infty} 2^{pk} \mu \{{x: |f(x)|>2^{k}}\} \lt \infty$ If $f \in L^p$, I applied the Chebyshev's inequality But ...
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### Is the inclusion map of $\ell^1(N)$ in $\ell^2(N)$ bounded and dense?

I am looking for an idea to prove if the inclusion map from $\ell^1(N)$ to $\ell^2(N)$ bounded and does it have a dense image. And why is the set $A:=\{x: ||x||_1\le 1\} \subset \ell^2(N)$ closed and ...
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### Closed subspace of $L^1[0,1]$

The statement I need to prove is following. Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ ...
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### Limit and Integral sign in $L^2$.

If $\lim_{k \to \infty} \| u_k - u \|_{L^2(\Bbb R^n)} = 0$ then how can I show that $$\lim_{k \to \infty} \int_{\Bbb R^n} u_k v = \int_{\Bbb R^n} uv$$ for any $v \in L^2 (\Bbb R^n)$?
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### Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q$, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
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### When multiplication operator is an bounded below/ open mapping/isometry/quotient map?

Let $(\Omega,\Sigma)$ be a $\sigma$-finite measurable space. Let $\mu,\nu\in \mathcal{M}(\Omega)$ be $\sigma$-additive measures on $\Omega$. Assume we are given $p,q\in[1,+\infty]$ and a measurable ...
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### Rademacher functions form an orthonormal system but not an orthonormal basis

I would like to know how to show that the functions $$r_n(t)=\operatorname{sgn}\big(\sin(2^n \pi t)\big)$$ (where $\operatorname{sgn}$ is the sign function) form an orthonormal system but not an ...
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### $L^2$ norm inequality

I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ ...
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### Negative integral on intervals implies negative function?

Let $f \in L^1([0,1])$ be such that for all $t \geq s$, $\displaystyle \int_s^t f(u)du \leq 0$. Is it true that $f\leq 0$ almost everywhere?
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### weak vs. norm compactness in $\ell_1$

So I'm trying to show that weakly compact sets in $\ell_1$ are norm-compact. I've already proven that weak sequential convergence implies norm convergence. I think the idea I want to go with is to ...
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### Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite ...
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### Dual space of the function $f$ in Fourier Transform

Let $f\in L^1{(\mathbb{R})}$. Why the Fourier Transform $\hat{f}\in L^{\infty}{(\mathbb{R})}$. Is it because $(L^1{(\mathbb{R})})^*=L^{\infty}{(\mathbb{R})}$?
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### $L_p$ Spaces and limits of translated functions

If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$ where $g_{(t)}(x):=g(t+x)$. Any hints? Try to give me only ...
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### Is the injection $\ell^p \subset \ell^q$ continuous for $p<q$?

It is easy to show that $\ell^p \subset \ell^q$ when $1 \leq p<q \leq + \infty$, but is the injection continuous? If so is $\ell^{\infty}$ the direct limit $\lim\limits_{\rightarrow} \ \ell^p$ as ...
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### What is the predual of $L^1$

Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a Banach space. How do you start to find such ...
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### Convergence in $L^{\infty}$ norm implies convergence in $L^1$ norm

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. Assume the measure space $X$ has finite measure. If $f_n$ converges to $f$ in $L^{\infty}$-...
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