# 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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### Relation between the types of convergences of sequences in $\mathbb{L}_p$ Spaces.

As far as I know there are 4 types of convergence in $\mathbb{L}_p$ spaces. 1. Pointwise Convergent a.e 2. Uniformly Convergent almost a.e 3. Convergence in Measure 4. p-Convergence My question is how ...
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### Existence of constant for a “Minkowski-like” inequality to hold on $L_p$ $p<1$.

I'm solving some problems to prepare for my phd qualifying exam on functional analysis and measure theory. I want to prove that given a measure space $(X,\mathcal{A},\mu)$ for every $0<p<\infty$...
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### multiplication of measurable functions in $L^p$ spaces

Let $(X, M, \mu)$ be a measure space, $q \in (0, +\infty]$ and $f,g : X \rightarrow \mathbb{C}$ in which $f \in L^{\infty} (\mu)$ and $g \in L^q (\mu)$. I want to show that $fg \in L^q (\mu)$. For ...
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### Show $l^p$ is not complete with the $q$ norm

I know the question has been asked here, but I do not understand the solution (Are $\ell_p$ spaces complete under the $q$-norm?) I came up with my own solution and was wondering if it is correct. ...
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### Understanding proof of Jensen's inequality in Lieb-Loss

I'm reading a book (Lieb–Loss) and in Section 2.2, they present a proof of Jensen's inequality and there's a step I don't quite understand. To set this up, suppose $J:\mathbb R\to\mathbb R$ is convex. ...
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### Prove the relative compactness of a sequence in $L^p$

Let $I=[0,1] \subseteq \mathbb{R}$ and let $(u_n),(v_n)$ be sequences in $C(I)$ such that $$|u_n(0)|+|v_n(0)| \le 1,~~~~ |u'_n(t)|+|v'_n(t)| \le t+e^t ~~~~ \forall t \in I, ~\forall n.$$ I would like ...
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### If $X_n$ converges to $X$ in $L_p$ and $Y_n$ converges to $Y$ in $L_p$ then $X_n + Y_n$ converges to $X + Y$ in $L_p$ [closed]

I want to show that if $X_n \xrightarrow{L^p} X$ and $Y_n \xrightarrow{L^p} Y$ then $X_n + Y_n \xrightarrow{L^p} X + Y$ ($p \geq 1)$. My idea is to use the following facts (whose proofs I won't give ...
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### Using a basis for $L^2(\mathbb{R}^d)$ to get a basis for $L^2(\{x\}\times\mathbb{R}^{d-k})$ for all $x$

I am in the situation where it would be very convenient if I could take a basis $\{f_i\}_{i = 1}^\infty$ for $L^2(\mathbb{R}^n)$ and manipulate it in some way to get a basis for the square-integrable ...
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### Why $L^2\cap L^p$ is not dense in $L^{\infty}$?

Ein Euclidean space $\mathbb{R}^n$. Why $L^2\cap L^p$ is not dense in $L^{\infty}$? I have that $L^2\cap L^p$ is dense in $L^p$ with $1\leq p<\infty.$ Indeed, for $g\in L^p$ with $1\leq p<\infty$...
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### Is $C_0$ dense in $l^{\infty}$

Is $C_0$ dense in $l^{p}$ with $1\leq p\leq \infty$ where $C_0=\{ (x_n): x_n\rightarrow 0, x_n\in R\}$. Well I think that if $p<\infty$ is true because by definition if i take $y=(y_n)\in l^p$ then ...
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### Topology of $p$-integrable functions space.

In a reference, I read that topology of $L^p(\mathbb{R}^n)$, $1\leq p\leq \infty$. What is the topology of $L^p(\mathbb{R}^n)$? I know that $f\in L^p$, $\|f\|_{p}^{p}=\int_{\mathbb{R}^n}|f(x)|^{p}dx$. ...
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### Extending an approximation result in $\mathbb{L}^p$ for $p<\infty$ to $\mathbb{L}^\infty$?

studying a proof for the $p-$independence of the $UMD_p-$ property for Banach spaces there was the following technical lemma. Lemma: Let $1\leq p<\infty$ and $\epsilon >0$ be given. If $f$ is a ...
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### Prove that there is a constant $M$ such that $\int|fg|dm\leq M \| f\|_{L^{p}}$ for all $f\in L^{p}(\mathbf{R})$.

I could not understand the last part of the proof of the following theorem: Let $p\geq 1$ and $g$ be a measurable function such that $\int|fg|dm<\infty$ for every $f\in L^{p}(\mathbf{R})$. ...
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### If $x \in \mathbb R^n$ with $\|x\|_\infty < 1$, how to make sense of this quantity $W(x):= \sum_{p=1}^\infty\|x\|_p^p$?

In some calculations of mine, I've stumbled on the following object, and I'm wondering if its a something recognisable. For $x \in \mathbb R^n$, let $\|x\|_\infty := \max_{1 \le j \le n}|x_j|$, and ...