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
19 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
3
votes
1answer
33 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
1answer
24 views

Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad ...
1
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0answers
23 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
1
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1answer
23 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
3
votes
1answer
30 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
2
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0answers
19 views

Continuity of Translation and Dilation on $L^p$ spaces

Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto ...
1
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1answer
22 views

Why can a function in $L^1(\partial \mathbb{D})$ be represented by a Fourier series?

I am looking for a reference to the claim that for any $f\in L^1(\partial \mathbb{D})$, where $\partial \mathbb{D}$ is the unit circle in $\mathbb{C}$, ...
1
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1answer
25 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
4
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1answer
25 views

Is $L^1_{loc}(\mathbb{R})$ complete with the norm $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy$

Let $BL^1_{loc}$ be the space of locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy<\infty$. Is this space complete ? What I tried: ...
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2answers
47 views

$L_p$ spaces and convergence in mesure [closed]

How can one prove the following one: Let $f,f_n\in \cal{L^p}$$(X)$, $\forall n\in\mathbb{N}$. If exists $g\in\cal{L^p}$$(X)$ such that $-g\le f_n\le g, \forall n\in{\mathbb{N}}$ and ...
3
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3answers
33 views

Inequality leading to Holder's: $t^\theta \leq \theta t + 1 - \theta$ with $0<\theta<1$ and $t\geq 0$

In the process of proving Holder's Inequality for $l^p$ spaces, as per my instructions it begins by first asking us to prove the following inequality as a first step: If $0<\theta<1$ and ...
3
votes
2answers
37 views

$\overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R)$

While reading a proof in a book they used the following result: $$ \overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R) $$ saying that it's well known !! But all I can see is ...
0
votes
1answer
25 views

Weak derivative and approximation

Let $f \in L^2(\mathbb R)$ be absolutely continuous. Is it true that if $f_n$ is a sequence of Lipschitz functions s.t. $f_n \rightarrow f$ in $L^2$ and $ f_n' \rightarrow g $ in $L^2$ then $ ...
2
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0answers
25 views

convergence in $L^p$ implies convergence in measure

I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in $L^p$ in measure, where $1\le p \le \infty$. Here is my attempt for $p>1$ - Let $\varepsilon>0$ and define ...
2
votes
1answer
41 views

Finding $p$ for which $f\in L^p(\mathbb{R}^2)$

Here's an old qual problem in analysis. Let $s=\Vert x \Vert$ and define $$f(x)=\frac{1}{s(1+s^{1/2}\log s)}$$, $x\in \mathbb{R}$. Find $1\le p\le \infty$ for which $f\in L^p(\mathbb{R}^2)$. Attempt ...
2
votes
1answer
80 views

If $g$ is the $L^2$-derivative of a function $f\in L^2$, then integrating $g$ gives $f$

If $f(x), g(x) \in L^2(\mathbb{R})$ and $\lim\limits_{h\to\infty}\int_{\mathbb{R}}|f_h(x)-g(x)|^2dx=0$, where $f_h(x):=\frac{f(x+h)-f(x)}{h}$ for any $h\neq 0$, show that ...
9
votes
4answers
100 views

What's so special about $p=2$ for the $L^p$ spaces?

The Banach space dual of $L^p$ is $L^q$, where $q=\frac{p}{p-1}$, but I don't really understand the motivation behind this. In particular, I find it kind of surprising that the only $L^p$ space whose ...
0
votes
2answers
44 views

A question about Fourier coefficients.

Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of ...
0
votes
1answer
11 views

Union and intersections of $L_p$ spaces and proper subsets.

Let $X= [0,1), S= \mathcal{B}_{[0,1)}, \lambda = $Lebesgue measure in $\mathcal{B}_{[0,1)} $ Prove (a) $L_p(\lambda) \subsetneq \bigcap_{0<r<p} L_r(\lambda) $ for every fixed p. (b) ...
1
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2answers
23 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 ...
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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
23 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\| = ...
2
votes
1answer
34 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
27 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 ...
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2answers
58 views

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

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
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1answer
34 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
44 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
68 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
25 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
53 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
62 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
52 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 $ ...
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0answers
27 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
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1answer
36 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 ...
4
votes
3answers
96 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
72 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
39 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
36 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
34 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
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
32 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
21 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
71 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
48 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
vote
2answers
33 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.
6
votes
1answer
45 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
40 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
22 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 ...