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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Exercise 7 page 93 Functional Analysis book of Conway

The following is Exercise 7 page 93 in Functional Analysis book of Conway: Let $1 \le p \le \infty$ and suppose $(a_{ij})$ is a matrix such that $(Af)(i) = \sum_{j=1}^{\infty} a_{ij} f(j)$ defines an element $Af$ of $\ell^p$ for every $f$ in…
user200918
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Riesz-Fischer in $L^\infty$ clarification needed.

This is Rudin's RCA book. He discusses completeness of $L^\infty(\mu)$, but there is a part of proof that I don't understand. In $L^\infty(\mu)$, suppose $\{f_n\}$ is a Cauchy sequence, and let $A_k, B_{m,n}$ be the sets where $|f_k(x)| >…
James C
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How does Holder's inequality apply to the expectation operator when using the infinity norm?

Let $F(x)$ be a CDF. Let $U_i$ ($1\le i \le n$) be i.i.d. random variates chosen from the standard uniform distribution on $(0,1)$. Let $U_{(i)}$ be the $i^{th}$ order statistic, i.e., $i$ of the $U$s are less than or equal to $U_{(i)}$. Let…
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Convergence of a series of functions in L^p

I found some problems in solving this exercise: Find $p>1$ such that the series $\sum_n f_n$ converges in $L^p(\mathbb{R})$ (with lebesgue…
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Same on dense subspace implies same on whole space?

I read the whole proof of Theorem 13.18 Bruckner's Real Analysis book and I had no problem understanding the proof except for following two claims inside the proof that are stated without further explanations. One is dense-ness of $L^{\infty}$ in…
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$L^p$ norm of $\frac{1}{1-\overline{\alpha }e^{i\theta}}.$

I am trying to calculate the $L^p$ norm of the function $f_\alpha(e^{i\theta}) = \frac{1}{1-\overline{\alpha }e^{i\theta}},$ where $\alpha$ is a complex number with $|\alpha| < 1.$ Id est, for every $1\leq p < \infty$ I would like to compute $$…
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If $\Omega\subseteq\mathbb R^d$ is sufficiently nice, does it hold $L^\infty(\Omega)\subseteq H^1(\Omega)$?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$ be bounded and open. Assume $\overline\Omega$ is a $d$-dimensional properly embedded $C^\infty$-submanifold of $\mathbb R^d$. Let$^1$ $(e_n)_{n\in\mathbb N}\subseteq H_0^1(\Omega)$ such that…
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Show that for $f(s,t)=(s^2+t)^{-a}$ the function $f^b$ is integrable over $(0,1)\times(0,T)$

This should be a rather simple question, but I wasn't able to figure out what's the trick to find a suitable upper bound: Given $a\in(1/2,3/2)$, $b\in[1,1/(a-1/2))$ and $T>0$, how can we show that for $f(s,t):=(s^2+t)^{-a}$, $s\in(0,1)$,…
0xbadf00d
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$f\in L^2(\mathbb{R})$ is absolutely continuous and $f' \in L^2(\mathbb{R})$ if and only if $\int |y\hat{f}(y)|^2dy<\infty$

This comes from Walter Rudin's Functional Analysis p. 388, exercise 21(c): But the domain I found on Engel's One-Parameter Semigroups for Linear Evolution Equations p. 66 is $$ D(A)=\{f\in L^2(\mathbb{R}):f \text{ is absolutely continuous, } f' \in…
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Is $\mathcal{B}(L^p,L^1)$ strictly bigger than $L^q$?

Let $(\Omega, \mathscr{F}, \mu)$ be a $\sigma$-finite measure space and $1\leq q \leq \infty$. Denote by $p$ the conjugate exponent of $q$. We know, given a $g\in L^q (\Omega)$, \begin{equation} T:f \in L^p(\Omega) \mapsto gf…
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A problem on weak convergence: weak convergence of a sequence implies the weak convergence of the square of this sequence?

Suppose that $\{u_m\}_m\subset H^1(\mathbb{R}^3)$ and $u_m\rightharpoonup u$ weakly in $H^1(\mathbb{R}^3)$. From the classic results, I know that there exists a subsequence such that $u_m\rightharpoonup u$ weakly in $L^2(\mathbb{R}^3)$. My question…
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About the proof of Minkowski generalized integral inequality

I am studying the generalized minkowski inequality for integrals: Let be $E\subset \mathbb{R^n}$, $F\subset \mathbb{R^m}$, $1\leq p \leq \infty$, $f$ measurable in $E\times F$. Then $$\left\| \displaystyle…
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Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, $F(t)=\int_{x_0}^{t}f(s)ds$. Prove that, $DF=f$ (towards theoretical distribution).

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, consider $$ F(t)=\int_{x_0}^{t}f(s)ds. $$ Prove that, $DF=f$ (towards theoretical distribution). I thought of the following: Let $\varphi$, we had $$ \left\langle DF, \varphi…
user915658
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Application of Closed Graph Theorem for l2 space

I have an exercise, which I am struggling with. Suppose $x\in \mathbb{R}^\mathbb{N}$ and $\forall y \in l^2$ we have $xy \in l^1$. Now I am to show that $x \in l^2$. There is a hint to use the Closed Graph Theorem. Now I have an idea, we define a…
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$f \in L_p(1,\infty ))$ if and only if $p \geq 2$

If $f(x)=\frac{1}{\sqrt{x} (1+\ln{x})}$ for $x \in (1, \infty)$. I need to show that $ f \in L^p(1, \infty))$ if and only if $p \geq 2$ I suppose that $p \geq 2$ and so $||f||_p^p=\int_1^{\infty}\frac{x^{-p/2}}{(1+\ln{x})^p}dx$ When I substitute…
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