# Tagged Questions

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### Integrable function with given condition is in $L^p$

Suppose $f:\Bbb R \to \Bbb R$ is integrable and there exist constant $c\gt 0$ and $\alpha \in (0,1)$ such $$\int_A |f(x)|dx\le cm(A)^\alpha$$ for every Borel measurable set $A\subset \Bbb R,$ where ...
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### What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
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### Showing a sequence is in $\ell^2$ [duplicate]

I am working on the following problem. Suppose that $\{a_j\}_{j=1}^{\infty}$ is a sequence with the property that, whenever $\{b_j\}_{j=1}^{\infty} \in \ell^2$, one has ...
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### Canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact?

Does there exist $q>p$ such that the canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact? My answer is no. Since we know that $L^\infty (0,1) \to L^p(0,1)$ is not compact, take $\{\sin(nx)\}$ ...
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### $\ell^p$ spaces' inclusion

$$\ell^s\subsetneq \bigcup_{k<p}\ell^k\subsetneq \ell^p\subsetneq\bigcap_{k>p}\ell^k\subseteq \ell^q$$ for any $1\le s<p<q$. Any idea to prove these inclusions? Counterexamples for the ...
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### Showing that a function is bounded in $L^1$ given a bound on its distribution function

Let $f \in L^2((0,T)\times\Omega)$ where $\Omega$ is a compact manifold. Suppose I know that for every $k > 0$, $$\mu(\{|f| > k\}) \leq Mk^{-\frac 12}$$ for some constant $M$ (which is ...
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### Need help with application of Hardy-Littlewood inequality (Marcinkiewicz space and distribution functions)

I am going over this work here. I couldn't understand the equality where the Hardy-Littlewood inequality is used. I think $\delta$ here is a weight so we can take it to be $1$ for simplicity. Would ...
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### Duality set for $L~p$ spaces, $1<p<\infty$.

I need to show that, given $f \in L^p$, $1<p<\infty$, the duality set $F(f)$ is equal to the point $$\|f\|_p^{2-p}|f|^{p-2}\overline{f}.$$ I have a hint: this is a consequence of convexity of ...
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### Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
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### $C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
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### Banach valued sequence spaces $\ell^p(X)$

Let $X$ be a Banach space and $\ell^p(X)$ denote the space of sequences $x_i\in X$ for which the norm $\big(\sum_{i=1}^\infty\|x_i\|^p\big)^\frac1p$ is finite, when $X=\mathbb{R}$ we get the usual ...
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### Given $u \in L^1$, is there approximating sequence $u_n \in L^\infty$ uniformly bounded in $L^p$?

Let $u \in L^1(U)$ where $U$ is a bounded domain. Is it possible to find a sequence $u_n \in L^\infty$ converging to $u$ in $L^1$ such that the $u_n$ are uniformly bounded for all $n$ in some $L^p$ ...
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### $L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
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### Convergence of Step Function Defined by Averages

For a function $f \in L^2[0,T]$, and a uniform partition $P = \{0=t_0, t_1, \ldots, t_n = T\}$ of the domain, we can define a step function approximation as the average value over each interval in the ...
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### p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
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### An application of Holder's inequality to show one norm is smaller than another

Let $p(s) = r(s) + m-1$ where $r:[0,T) \to [q,\infty)$ where $q \geq 2$ and $m > 1$ is fixed. Let $\text{Vol}(\Omega) = 1$. Then can we show that \lVert u \rVert_{L^{r(s)}(\Omega)} \leq ...
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### Proving that weak limit in $L^p$ and strong limit in $H^{-1}$ are the same

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. Let $p \geq 1$ and suppose that $u_n \rightharpoonup u$ in $L^p(\Omega)$ and $u_n \to v$ in $H^{-1}(\Omega)$. How to show that $u=v$? I can do ...