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 ...

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1answer
30 views

Rudin Real & Complex Analysis Thm 3.14

In the proof, the author claims that by Lusin's theorem, $g(x) = s(x)$ except on a set of measure $< \epsilon$ and $|g| \leq \|s\|_\infty$, ($g(x) \in C_c(X)$, $s(x)$ simple and $\mu(\{x:s(x) \neq ...
5
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3answers
85 views

The function $\phi(p)=\|f\|_{L^p}^p$ is convex

Fix an arbitrary function $f\in L^p([0,1])$ and define $$\phi(p)=\|f\|_{L^p}^p$$ for $p\in [1,\infty)$. Prove $\phi$ is convex. Comments: This is a standard property of $L^p$ spaces, but no ...
2
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2answers
59 views

The $L^p(\mathbb R)$ norm is increasing as a function of $p$ (Update: It's false!)

Update: This is false. See the answers for a counterexample. Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p \right)^{1/p}$$ is ...
3
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1answer
33 views

Intuition behind the Riesz-Thorin Interpolation Theorem

Quoting the definition on Wikipedia, Let $(\Omega_1, \Sigma_1, \mu_1)$ and $(\Omega_2, \Sigma_2, \mu_2)$ be $\sigma$-finite measure spaces. Suppose $1 \leq p_0 \leq p_1 \leq \infty$, $1 \leq q_0 ...
1
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0answers
31 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
2
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2answers
29 views

Why is the zero extension of an $L^p$ function in $L^p$?

Let $u \in L^p(0,1)$. Define $\tilde u:(0,\infty) \to \mathbb{R}$ as the function which equals $u$ on $(0,1)$ and $\tilde u =0$ on $(1,\infty)$. I cannot figure out why this function is measurable. ...
4
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1answer
23 views

On showing that if $f_n \to f, g_n \to g$ in $L^p$ then $max(f_n, g_n) \to max(f, g)$ in $L^p$

Let $(f_n)$ and $(g_n)$ be two sequences in $L^p(\Omega)$ with $1 \leq p < \infty$ such that $f_n \to f$ in $L^p(\Omega)$ and $g_n \to g$ in $L^p(\Omega)$. Let $h_n = max(f_n, g_n)$ and $h = max(f, ...
1
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2answers
30 views

Prove that a set is orthonormal on $L_2$

I would like to prove that the set of elements: \begin{equation} A_n(t)=\left\{\frac{1}{\sqrt{2\pi}}e^{int}\right\}_{n=-\infty}^{\infty} \end{equation} is an infinite orthonormal set, on space ...
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0answers
37 views

How can I prove that [on hold]

Let $μ^*$ be an outer measure on $P(X)$ and $m^*$ be the class of measurable sets with respect to the outer measure $μ^*$. How can I prove that the class of measurable set $m^*$ with respect to the ...
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1answer
29 views

The Essential Supremum as a Limit

Let $(X, \mathcal F, \mu)$ be a finite measure space and let $f\in L^\infty(X, \mu)$. Define $\alpha_n=\int_X |f|^n\ d\mu$. Then $$\lim_{n\to \infty}\frac{\alpha_{n+1}}{\alpha_n}=\|f\|_\infty$$ ...
2
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0answers
18 views

Prove $\mathcal{L}_p[0,1]$ is separable using Lusin and Stone-Weierstrass theorems.

1. Prove that the set of $p$-integrable functions on $[0,1]$ with the Lebesgue measure $\lambda$ is separable using Lusin's Theorem and the Stone Weierstrass Theorem. 2. Prove that ...
2
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1answer
68 views

Showing that $\int fg\le \int g$ implies $f=0$ a.e.

Take $0<p<1$. If $f$ is locally integrable over on $\mathbb{R}$ and $$\Bigg\vert \int fg\Bigg\vert\le \Vert g\Vert_p\tag 1$$ for every $g$ continuous on a set of compact support, then $f=0$ a.e. ...
1
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0answers
43 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
1
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2answers
31 views

$L^p$ spaces and proper inclusion

Let $1≤p < q$. Prove that $L^p(\mathbb{R}) \subset L^q(\mathbb{R})$ and the inclusion is proper. I am unsure how to begin this or even prove it about $L^p$ spaces and Banach spaces.
5
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1answer
42 views

Inclusions of $\ell^p$ and $L^p$ spaces

I remember seeing this some time ago, but I can't find the examples anywhere. Recall that if $p<q$, then $\ell^p\subseteq\ell^q$ and $L^q[0,1]\subseteq L^p[0,1]$. So we can ask ourselves if any of ...
0
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2answers
37 views

$f_1,f_2 \in L^q(\mu)$ and $\int_\mathcal{X}f_1gd\mu = \int_\mathcal{X}f_2gd\mu$ for all $g \in L^p(\mu)$ implies $f_1=f_2$ a.e.

Let $X=(\mathcal{X},\mathcal{M},\mu) $ be a measure space. Assume that $\mu$ is $\sigma$ finite and $1\leq p \leq \infty$, with $q$ the Holder conjugate exponent. If $f_1,f_2 \in L^q(\mu)$ and ...
1
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0answers
18 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
0
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1answer
30 views

sequence in $L^1$ converging pointwise a.e., but not weakly.

Find a $(X,\Sigma,\mu)$, a $\sigma$-finite measure space and a norm-bounded sequence $\{f_n\}$ in $L^1(X)$ that converges almost everywhere to $f$ but does not converge weakly to $f$. Can you help me ...
0
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1answer
27 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
2
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1answer
23 views

Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$. Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) ...
1
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1answer
39 views

Finite meaure space with $f \in L^p$ [duplicate]

Given a finite measure space $(X,\Sigma,\mu)$, for $1<p<\infty$, if $f \in L^p(X)$, then $f \in L^1(X)$. Can anyone show me how to start the proof? Thanks.
1
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1answer
25 views

Any function in $L^p$ space is a linear combination of simple functions? True OR not?

Any function in $L^p$ space is a linear combination of simple functions for $1<p<\infty$. Is this true? So any function in $L^p$ is measurable. So any measurable function can be represented ...
1
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2answers
58 views

Is a subspace of functions that essentially depend only on one variable closed?

Let $S$ be the subspace $$\left\{f\in L^p( I^2)|\exists g\in L^p( I), f(x,y)=g(x), \mbox{a.e. } (x,y)\in I^2\right\}.$$ Is $S$ closed under the $L^p$ norm? I think the first step would be to ...
4
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1answer
60 views

Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant. Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< ...
0
votes
1answer
29 views

Showing that a function is L1

I have been struggling with this problem; it should just use some basic inequalities, but having difficulty getting them in the right order. Let $f \in L^2(\mathbb{R})$ such that it is also the case ...
4
votes
1answer
101 views

How to check some topological concepts in product and direct sum spaces

Given $a=(a_i)_{i=1}^\infty$ with $a_i \geq 0$ and $b=(b_i)_{i=1}^\infty$ with $b_i \in \mathbb{R}$, let $$E_i = \lbrace (x_n)_{n=1}^\infty : n^{b_i}|x_n|\leq a_i, \forall n\in \mathbb{N} \rbrace$$ ...
1
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1answer
25 views

Proving a function is L2

I have been trying to solve the following problem using Holder's or Young's Inequality, but I am just not doing the right manipulations I think. It shouldn't require anything fancy. Let $f \in ...
2
votes
2answers
36 views

$f \in L^p(\mathbb{R^d})\cap L^r(\mathbb{R^d})$ implies $f\in L^q(\mathbb{R^d})$ where $p<q<r$

Prove that if $f \in L^p(\mathbb{R^d})\cap L^r(\mathbb{R^d})$ then $f\in L^q(\mathbb{R^d})$ where $p<q<r$. I have tried all the usual $L^p$ inequalities but dead end every time. I think I ...
2
votes
0answers
29 views

Inequality for $L^p$ spaces

Prove that $\|f+g\|_4^4+\|f-g\|_4^4 \leq (\|f\|_4+\|g\|_4)^4 + (\|f\|_4-\|g\|_4)^4$ for $f,g :\mathbb{R}^d \rightarrow \mathbb{R}$. Notice that you can't solve it with minkowski's inequality. My ...
1
vote
1answer
34 views

How can I prove Holder Ineqaulity for $0<p<1$

$0<p<1$ $\dfrac{1}{p}+\dfrac{1}{p'}=1$ if , $f \in L^{p}$ and $0<\int_{\Omega}\vert g(x) \vert^{p'}dx < \infty$ then $$\int_{\Omega}\vert f(x)g(x) \vert dx \geq (\int_{\Omega}\vert ...
0
votes
1answer
28 views

Duality (conjugate) function for $f \in L^\infty$

For $f\in L^\infty(\mathbb{R})$, can I find the $f^*\in (L^\infty(\mathbb{R}))^*$ such that $$\|f^*\|_* = 1 \text{ and } \langle f^*, f \rangle = \|f\|_\infty.$$ I know that ...
6
votes
1answer
115 views

Generalization of Minkowski inequality

I am wondering if the following is true: Suppose continuous function $g: [0, \infty) \to [0, \infty)$ satisfying $g(0)=0$ is increasing and strictly convex and (therefore) invertible. Let $||f ...
0
votes
1answer
35 views

Convergence in $L^2$ norm given inner product converges

Suppose $(f_n)$ and $f$ are in $L^2$ and $\int_E f_ng \rightarrow \int_E fg,\,$ for $g \in L^2(E)$. If $\|\,f_n\|_2 \rightarrow \|\,f\|_2$, then show that $f_n \rightarrow f$ in $L^2$. (All functions ...
1
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1answer
39 views

Hölder's inequality, equality condition

It is known that equality in Holder's inequality (i.e $|\int_Efg|=||f||_p||g||_q$) holds iff $\|g\|_q^q|f|^p=\|f\|_p^p|g|^q$ a.e. However recently I read somewhere that an additional constraint must ...
0
votes
1answer
27 views

Definition of Strong Convergence in $L^p$

Is strong convergence in $L^p$, ie $f_i \overset{strongly}\longrightarrow$, just $||f_i-f||_{L^p} \rightarrow 0$. If so why dont we just call it convergence?
2
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2answers
35 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwartz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
3
votes
0answers
121 views

Dense subset of $L^p[a,b]$

It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ? ...
2
votes
2answers
28 views

Show that a subspace is proper dense in $l^1$ sequence space. L^1 space.

Let Y = $L^1 $($\mu$) where $\mu$ is counting measure on N. Let X = {$f$ $\in$ Y : $\sum_{n=1}^{\infty}$ n|$f(n)$|<$\infty$}, equipped with the $L^1$ norm. Show that X is a proper dense subspace ...
2
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0answers
112 views

Characterization of $L^p$ function for $p>1$

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, $1<p<\infty, 1/p +1/q=1,f \in L^1(\Omega)$. Is it true that if there exists $C>0$ such that $sup \{\int fg: g$ is a step function, $ ...
2
votes
2answers
47 views

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?
2
votes
1answer
46 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
0
votes
1answer
14 views

Does $\partial_b u(\cdot,b) \in L^2(A)$ for fixed $b$ imply that $u(\cdot,b) \in L^2(A)$?

Let $u$ be defined on $A \times B$ where $A$ and $B$ are two bounded domains, and write $u=u(a,b)$. Suppose that the weak derivative $\partial_b u(\cdot,b) \in L^2(A)$ for fixed $b$. Does this imply ...
4
votes
1answer
38 views

$\||f|^{2}f\|_{H^{s}}\leq C \|f\|^{2}_{L^{\infty}} \|f\|_{H^{s}}$ for $s>0, f\in H^{s}(\mathbb R)$?

We let $H^{s}(\mathbb R^{n}), (s\in \mathbb R)$ the Sobolev spaces. It is well known that: the space $H^{s}(\mathbb R^{n})$ is an algebra with respect to pointwise multiplications, for $s>n/2.$ ...
2
votes
1answer
65 views

Prove that $T$ is bounded if $\langle Tx, y \rangle = \langle x, T^*y \rangle$

Suppose $T: L^2(\mathbb R^d) \to L^2(\mathbb R^d)$ is a linear operator, and there exists $T^*: L^2(\mathbb R^d) \to L^2(\mathbb R^d)$ such that $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for ...
2
votes
3answers
51 views

$\|fg\|_{L^2(\Omega)}\leq \|f\|_{L^2(\Omega)}\|g\|_{L^\infty(\Omega)}$

is true that $\|fg\|_{L^2(\Omega)}\leq \|f\|_{L^2(\Omega)}\|g\|_{L^{\color{blue}\infty}(\Omega)}$ ? I can't see a proof for this :/ ( of course, $\|fg\|_{L^2(\Omega)}\leq ...
2
votes
1answer
44 views

Find all functions such that $ (\int _0 ^1 xf(x) dx)^3 = \frac{4}{25} \int _0 ^1 f(x)^3 dx$

Calculate all the functions $f \in L^3$ such that $$ \left(\int _0 ^1 xf(x) dx\right)^3 = \frac{4}{25} \int _0 ^1 f(x)^3 dx$$ Can someone please walk me through this because there are no such ...
2
votes
1answer
20 views

Convergence of a product of sequences convergent in mean when one of them is bounded

Suppose $X_n\to X$ in $L^1$ and $V_n\to V$ in $L^1$ and $(V_n)$ is a bounded sequence. I'm trying to show that then $\mathbb{E}X_nV_n\to \mathbb{E}XV$. One has for all $N\in\mathbb{N}$ ...
0
votes
2answers
40 views

$\int_{\mathbb R}|f(x)|^{2} dx <\infty \implies \sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx <\infty$?

Let $f\in L^{2}(\mathbb R),$ that is, $\int_{\mathbb R}|f(x)|^{2} dx <\infty,$ and $\beta>0.$ My Question: Is it true that that: $\sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx ...
3
votes
1answer
42 views

Convergence in $L^1_{loc}$ implies convergence almost everywhere

Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ ...
1
vote
1answer
39 views

is $L^2 (\mathbb R)\subset L^\infty(\mathbb R)$?

I know that because i'm working on an infinite-measure space it could be tricky. And from my experience the answer to my question is probably no.. But nevertheless, I can't think of a non-bounded ...