# 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.

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### Show that if the integral of function with compact support on straight line is zero, then $f$ is zero almost everywhere

I want to prove that that given $f:R^2 \rightarrow R$ which is continuous with compact support s.t the integral of $f$ for every straight line $l$ is zero ($\int f(l(t))\mathrm{d}t=0$) then $f$ is ...
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### Is the space of all convergent sequences compact in the space of all bounded sequences($l^{\infty})$?

Is the space of all convergent sequences compact in the space of all bounded sequences($l^{\infty}$)? Argument: I think the answer is in yes because in particular if we consider a sequence of all ...
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### Union of $l^p$, 0<p<1

Is it true that $$\bigcup\limits_{0<p<1} l^p = l^1\quad?$$ The space $l^p$ is the space of the sequences $\{a_n\}_n$ with $\sum |a_n|^p <\infty.$ The one inclusion is obvious, as any ...
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### Example of $(L^1)^* \neq L^\infty$ from Exercise 6.12 in Rudin's RCA

This is Exercise 6.12 from Rudin's RCA. Let $\mathscr{M}$ be the collection of all sets $E$ in the unit interval $[0,1]$ such that either $E$ or its complement is at most countable. Let $\mu$ be the ...
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### Why is $(e_n)$ not a basis for $\ell_\infty$?

Let $(e_n)$ (where $e_n$ has a 1 in the $n$-th place and zeros otherwise) be unit standard vectors of $\ell_\infty$. Why is $(e_n)$ not a basis for $\ell_\infty$? Thanks.
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### On convergence in $L^p$

I was trying to understand an exercise about convergence in $L^p$. I was asked to investigate the punctual convergence and the $L^p(\Bbb R)$ convergence, $1\le p\le +\infty$, of $u_n(x)=1/n*e^{-|x|/n}$...
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### Use Fatou Lemma to show that $f$ takes real values almost everywhere.

Let $(f_n)$ be a sequence in $L^p$ such that for each positive integer $n$, $\| f_{n+1}-f_n\|_{p} <\frac 1{2^n}$. Define $f: X \to [0,\infty]$ with  f(x)= \sum_{n=1}^\infty| f_{n+1}(x)-...
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### Understanding a step in a proof of $L^p(X)\subseteq L^r(X)\subseteq L^1(X)$

I am trying to understand this proof to the question: $L^p$ and $L^q$ space inclusion. Here is the linked answer I am reading: There is a easy way to show that. Suppose that $p<q$ and X a space ...
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If $\Omega$ is a space of finite measure, then it is well known that $L^{N}(\Omega) \subseteq L^{2}(\Omega)$ for $N \in [2, \infty]$. I want to know if the image of this embedding is closed in \left(... 3answers 69 views ### How to prove the “bang-bang” characterization on the alignment betweenL_1[0,1]$and$L_\infty[0,1]\$?

I am following Luenberger's book "Optimization by Vector Space Methods". The author's solution to Example 2 on Page 124 obviously has utilized the following result on the characteristic of alignment ...