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.

$L^p$ spaces are defined for $p\in(0,\infty]$ as follows. Let $(X,\mathcal F,\mu)$ be a measure space. For $p$ with $0 < p<\infty$ we write $L^p(X,\mu)$ ($L^p(X)$ or $L^p$ when there is no ambiguity), for the vector space of equivalence classes (for equality almost everywhere) of measurable functions $f$ such that $\int_X|f|^pd\mu$ is finite. When $1\leq p<\infty$ we endow $L^p$ with the norm $\lVert f\rVert_p:=\left(\int_X|f(x)|^pd\mu(x)\right)^{1/p}$, while for $0 < p < 1$ we write $\lVert f\rVert_p:=\int_X|f(x)|^pd\mu(x)$, which induces a metric on $L^p$.

For $p=+\infty$, $L^\infty$ is the space of equivalence classes of functions $f$ such that we can find a constant $C$ with $|f(x)|\leq C$ almost everywhere. Then $\lVert f\rVert_{\infty}$ is the infimum of constants $C$ satisfying the latter property, and is called the essential supremum of $|f|$.

These spaces are sometimes called Lebesgue spaces. If we work with counting measure on, for example, the set $\mathbb N$, we get sequence spaces $\ell^p$ and $\ell^\infty$ as special cases.

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If $V\subset L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$, then $V$ is finite dimensional

If $V$ is a linear subspace of $L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$ for all $f\in V$, then $V$ is finite dimensional. The proof is an explicit calculation: Since $L^\infty[0,1] \subset L^2[0,1]$, take $e_1,\cdots , e_n$ to be…
Xiao
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In $\ell^p$, if an operator commutes with left shift, it is continuous?

Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution. What we have: The left shift $L : \ell^p \to…
Dario
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$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
user97601
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When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?

I would like to see a proof of when equality holds in Minkowski's inequality. Minkowski's inequality. If $1\le p<\infty$ and $f,g\in L^p$, then $$\|f+g\|_p \le \|f\|_p + \|g\|_p.$$ The proof is quite different for when $p=1$ and when $1
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How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, where $p > 1$, $p \neq 2$. Is it correct that $E$ and…
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Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by the following:$$\lim_{h\to 0} \frac{…
Giuseppe Negro
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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 almost everywhere $0.$ Well I know how to proof it…
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Distance minimizers in $L^1$ and $L^{\infty}$

If $H$ is a Hilbert space, we have the Hilbert Projection Theorem, which tells us that given a nonempty, closed, convex subset $K \subset H$, and a point $x \in H$, there is a unique point $y \in K$ which minimizes $\lVert x-y \rVert$. In the…
Elchanan Solomon
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$L^{\infty}$ is a Banach Space

Show that $L^{\infty}$ is a Banach Space with respect to the norm $||.||_{\infty}$ where $||f||_{\infty}=\inf\{ a\ge 0: \mu\left(\{x: |f(x)| \gt a\}\right)=0\}$ I am going to prove the following lemmas and then use them in the proof. Lemma-1:…
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Smooth functions with compact support are dense in $L^1$

Here is another homework question that I did and I'd be glad if you could tell me if it's right. We now strengthen the result of Question Two for $R$ where we have the notion of differentiability. Prove that for any open $Ω ⊂ R$ the set of smooth…
Rudy the Reindeer
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Completeness of $\{ f_n : n \in \mathbb N \} \subset C[0,1]$ in $L^1[0,1]$

Suppose that $\{ f_n : n \in \mathbb N \} \subset C[0,1]$ is a system of continuous functions which is complete in $L^1[0,1]$, i.e. each $g \in L^1[0,1]$ can be approximated arbitrarily well by a finite linear combination of the $f_n$'s. Is it true…
Muzi
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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 $\alpha$. We are given an $f \in L^p$ with $1\leqslant p <…
nullUser
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A Hamel basis for $\ell^p$?

I am looking for an explicit example for a Hamel basis for $\ell^{p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one can express any element of the vector space as a…
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Inclusion of $l^p$ space for sequences

Inclusion of $L^p$ spaces for functions has been discussed here. Does this apply to $l^p$ space of sequences similarly? I tried to show the following: For $1\leq p
Spock
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