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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1
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2answers
16 views

If $f \in L_4([0,1])$ then $f \in L_2([0,1])$ and $||f||_2 \leq ||f||_4$

If $f \in L_4([0,1])$ then $f \in L_2([0,1])$ and $||f||_2 \leq ||f||_4$ I am not sure how to prove the first statement, we say that $f \in L_P$ if $\int |f|^p < \infty$. Then if $f \in ...
-2
votes
0answers
39 views

A problem about real analysis [duplicate]

Show $\int_\mathbb{R^n} \left|f(x+h)-f(x)\right|^p dx \to 0$ as $h\to 0$, $f\in L^p (\mathbb{R^n}).$
3
votes
2answers
20 views

Showing that $\lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p$

I am reviewing a proof about the completeness of $L^p$ spaces. The proof begins as such (Folland Theorem 6.6): For $1 \le p < \infty$, suppose $\{f_k\} \subset L^p$ and $\sum_1^\infty \|f_k\| = ...
1
vote
1answer
32 views

The infinity norm of the sequence $v(n) = n \sin(n!)/(n^2+1)$

For a bounded sequence $v(n)$, $n\in\mathbb{Z}$ define $$||v||_\infty = \max_{n\in\mathbb{Z}} |v(n)|.$$ Let $$v(n) =\frac{n\sin{(n!)}}{n^2+1},$$ and find whether $||v||_\infty<\infty$. ...
4
votes
0answers
20 views

Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.

Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure. My work: I proved the ...
-1
votes
2answers
46 views

Does this hold for $p=\infty $, i.e., is it true that $(l^{\infty})'= l^1? $ [on hold]

Let $E=l^p$ where $1 \le p < \infty $ we know $E'=l^q$ Where $q$ is the dual exponent of $p$, i.e. $q$ is such that $\frac{1}{p}+\frac{1}{q}=1$ Does this hold for $p=\infty $, i.e., is it true ...
1
vote
1answer
32 views

Showing $f$ is integrable on a plane, given a bound on its $L^{3/2}$ norm on certain regions

(old qual question in analysis) If $A_\lambda=\lbrace (x,y): \lambda \le x^4+y^2\le 2\lambda \rbrace$ and $f$ is locally in $L^{(3/2)}(\mathbb{R}^2)$ and there is an $a>3/8$, such that ...
1
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0answers
42 views

Proving that the dual of $\ell^p$ is $\ell^\infty$ for $0<p<1$.

This question comes from Rudin's Functional Analysis, exercise 3.5(d). It concerns the $\ell^p$ spaces (for $0<p<1$) topologized by the metric $d(x,y)=\sum_{k=1}^\infty ...
1
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0answers
66 views

Holder Inequality when $0 < p < 1$ [duplicate]

If $0 < p < 1$, $f \in L^p$, and $\int \lvert g \rvert^q < \infty$, show that $$\int \lvert fg \rvert \ge (\int \lvert f \rvert^p)^{\frac{1}{p}}(\int \lvert g \rvert^q)^{\frac{1}{q}}$$ My ...
0
votes
1answer
23 views

How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality?

Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u ...
1
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2answers
49 views

Why is it important that $L^P$ spaces be complete?

I know that Banach spaces are ubiquitous and incredibly important in a lot of areas of math, but I was hoping for an intuitive explanation as to why (and when) it's important in the case of $L^p$ ...
3
votes
2answers
61 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
2
votes
2answers
48 views

Can $ {L^{1}}(G) $ be a $ C^{*} $-algebra?

Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ ...
1
vote
0answers
24 views

Need help understanding this proof of a certain inequality of $L^p$ norms.

The following theorem and proof is lifted from Folland (Real Analysis: Modern Techniques and their Applications). I am having trouble understanding one single line of the proof: Theorem: Let $K$ be a ...
1
vote
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
votes
3answers
92 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
votes
2answers
67 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
votes
1answer
36 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
vote
0answers
34 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
votes
2answers
31 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
votes
1answer
25 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
vote
2answers
31 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 ...
1
vote
1answer
30 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
votes
0answers
19 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
votes
1answer
70 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
vote
0answers
44 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
32 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
votes
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
votes
2answers
39 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
vote
0answers
21 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
votes
1answer
34 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
votes
1answer
28 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
votes
1answer
24 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
vote
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
vote
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
vote
2answers
61 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
votes
1answer
62 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
31 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
104 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
vote
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
40 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
1answer
32 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
47 views

How can I prove Holder inequality for $0<p<1$? [on hold]

$0<p<1$ $\dfrac{1}{p}+\dfrac{1}{p'}=1$ 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 f(x) ...
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
116 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
37 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
vote
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
29 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
votes
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
126 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 $1\leq p< \infty$ ? ...