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
52 views

Minkowski-like integral inequality

Let $f\colon\Omega\times\Omega\to\mathbb{R}$ be non-negative. Under which hypotheses does the following inequality hold $$ \left\{ \int \left[ \int f(x,y)\; \mathrm{d}\mu(y) \right]^p ...
0
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1answer
42 views

Boundedness of continuous summable function

Let $f\colon\mathbb{R}\to\mathbb{C}$ be a continuous function. If we suppose that $f$ is a $L^1(\mathbb{R;C})$ function too, then can we conclude that $f$ is bounded? ADD: I asked the preceding ...
3
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1answer
72 views

Find a function in $H^{\frac{1}{2}}$ that is not in $L^{\infty}$.

Les $\mathcal{S}$ be the Schwartz class and $\mathcal{S}'$ be its dual (also known as the set of tempered distributions). For a function $u$ let $\hat{u}$ denote de Frourier transform of $u$. Given a ...
1
vote
1answer
94 views

A question about Riesz - Fischer theorem's proof

In Riesz-Fischer theorem's proof, when we put $$ g_k =|f_{n_1}|+|f_{n_2}-f_{n_1}|+ \cdots + |f_{n_k}-f_{n_{k-1}}| $$ it is easy to get (by Minkowski's inequality) $$ \left \| g_k \right \|_p \leq ...
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0answers
47 views

A problem with a proof that $L^{p_{2}}\subseteq L^{p_{1}}$ for $1\leq p_{1}\leq p_{2}\leq\infty$

In a functional analysis course I saw a claim that for $1\leq p_{1}\leq p_{2}\leq\infty$we have it that $L^{p_{2}}\subseteq L^{p_{1}}$ I have a few problems with the proof given, and I would ...
0
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1answer
53 views

Examples of nonconvergent Cauchy sequences of functions

Let $f$ be a continuous, real valued function defined on a closed, bounded interval $I$, a subset of real numbers. Let $\{f_n\}$ be a Cauchy sequence in the $L^2$ norm. Give a counterexample that the ...
3
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1answer
91 views

Measure spaces are proper subsets

I want to prove that $L^2$ is of the first category in $L^1$, thus I have to prove that $L^2$ is the countable union of nowhere dense subsets. The hint I get is: Take $g_n(x)=n$ for $0\leq ...
2
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0answers
36 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
1
vote
1answer
406 views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
3
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1answer
68 views

Is there an easy/elementary way to show that if $f_k\to f \in L^p(\Omega)$ we can conclude that $\int |f_k|^p\to \int |f|^p$

Hey just stumbled upon this problem. Often one finds the Lebesgue dominated convergence theorem presented for $L^1$ spaces with the statement that the integrals do converge, i.e. $\int f_k \to \int ...
5
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0answers
118 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
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0answers
85 views

Orthonormal basis for this $L_2$ space of probability measures?

I couldn't find this on math.se or by searching the internet. Thanks for any help! OK, so I have a set $\Omega$. For now, let's think of $\Omega$ as finite, consisting of $n$ elements. Now consider ...
2
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1answer
65 views

Function bounded a. e.

I have a question: if $f$ is uniformly bounded in $L^2(0,T,X)$ , then $f$ is uniformly bounded a.e. in $X \times (0,T).$ If yes, how to prove it? Thank you.
2
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0answers
43 views

How to determine the limit?

How to solve following: Let $p\geq 1$ and $f\in L^p[0,1]$. For $\alpha\in\mathbb{R}$ determine $\lim_{x\rightarrow 0} x^\alpha\int_0^x f(t) dt$. This problem probably has several cases, but I'm ...
1
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1answer
46 views

Convergence in $L^p$ of an approximation to a function

Suppose you have a function $f\in L^p(\mathbb{R}^n)$ and some bounded set $\Omega$ of measure 1. Define $$ f_\epsilon = \frac1{\epsilon^n}\int_{\Omega_\epsilon} f(x + y)dy $$ Where $\Omega_\epsilon$ ...
0
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2answers
118 views

A relative compactness criterion in $\ell^p$

There is a relative compactness criterion for subset of $\ell^p$ that seems to me to be almost unheard-of (I say that because a google search provided no proofs nor references) but that is very ...
1
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1answer
91 views

Find functions in $L^2([0,2\pi])$ such that are orthogonal to $\sin(nx)$ and $\cos(nx)$

I have to find functions $u \in L^2([0,2\pi])$ such that these two hold: $$ \int_0^{2\pi} u(x)\sin(nx) dx = 0 \qquad \int_0^{2\pi} u(x)\cos(nx) dx = 0$$ for any $n = 1,2,3,...$ I tried to argue ...
3
votes
2answers
102 views

Geometric Mean limit of $\ell_p$ norm of sums

My analysis professor introduced the $\ell_p$ norm to our class as: \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} We are asked to prove the following: ...
1
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0answers
57 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...
1
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0answers
69 views

Definition of $L^p(\mathbb T)$ with $\mathbb T$ the unit circle

I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation ...
2
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1answer
94 views

How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..

can anyone help me show that if $f\in L^p(a, b)$ then $$ \lim_{t\to 0^{+}}\int_{a}^b|f(x+t)-f(x)|^p\ dx=0.$$ Thanks, any help will be useful..
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0answers
128 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
1
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1answer
48 views

Weakest conditions on a function so that integral is bounded

Let $\Omega$ be a bounded set and let $f:\Omega \to \mathbb{R}$. What are the weakest conditions of $f$ so that $$\int_{\Omega}uvf \leq C\lVert u \rVert\lVert v\rVert$$ holds for all $u, v\in ...
1
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1answer
55 views

Some properties about $L^p$ with $0<p<1$

We are coming across many Banach spaces $L^p$ with $1\leq p\leq\infty$. But how about $0<p<1$? Can it be normed? How about its metric induced by the norm? And how about its ...
1
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1answer
30 views

Looking for a basis of $L^2$ with this special property

The setup. Let $\mathbb{T^2}$ denote the two-dimensional torus, i.e. $$ \mathbb{T}^2 \simeq [-\pi,\pi)^2 $$ induced by identifying opposing faces of $[-\pi,\pi)^2$. Note that $$ L^2(\mathbb{T^2}) ...
2
votes
1answer
63 views

Integral on $\ell^{\infty}$

I begin with measure and integral theory. I want to give answer on the following statement: Suppose $l^{\infty}$ is the Rieszspace of all bounded functions on $\mathbb{N}$. Define ...
2
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0answers
44 views

A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
1
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0answers
43 views

How to show that entries of this matrix are in $L^\infty(0,T)$?

I have a problem. Let $A(t)$ be a $n \times n$ matrix for each $t \in [0,b]$ with the property for all vectors $x$ that $$x^TA(t)x \geq C|x|^2$$ where $C$ doesn't depend on $t$. Can I use this fact ...
6
votes
2answers
238 views

Subspaces of $L^p$

So studying Qualifying Exam problems in Analysis I cam across this one: For $1\lt r \lt p \lt s \lt \infty$ where $\mu$ denotes Lebesgue measure, a) Construct a subspace of $L^p([0,1],\mu)$ such ...
1
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1answer
92 views

If the weighted Lp norm of a measurable function is finite, is the weighted Lp norm of the antiderivative also finite?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} dx < \infty. $$ Define $g : \mathbb{R} \rightarrow \mathbb{R}$ to be $$ ...
3
votes
2answers
61 views

Don't understand this $L^p$ space inequality (Bochner spaces, etc)

For $p \geq 1,$ define $f \in L^p(0,T;X)$ by $$f = \sum_{i=1}^\infty x_i h \chi_{E_i}$$ where $E_i$ are measurable disjoint partition of $[0,T]$. The $x_i \in X$ with $|x_i|_X = 1$, and $h \geq 0$ is ...
1
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2answers
85 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
2
votes
2answers
108 views

When is the logarithm of this function square integrable?

I was trying to prove $\log(1-|r(s)|^2)$ lies in $L^2$ when $r\in L^1\cap L^2$. How should I do this? Thank you!
2
votes
0answers
43 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
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1answer
185 views

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like All classical (separable) Banach spaces used in analysis have a Schauder basis . I was wondering where to find a proof confirming ...
2
votes
2answers
80 views

$L^p$ norm representation via duality

I know that in a $\sigma$-finite measure space if $1\le p < \infty$ then $(L^p)'\cong L^{p'}$ isometrically via the isomorphism $$L^{p'} \ni g\mapsto \int \cdot\, g\ \mathrm{d}\mu \in (L^p)'.$$ ...
1
vote
1answer
114 views

Limit of consecutive Lp norms [duplicate]

I've been wrestling with the following proof off and on for a number of days, and I'm in need of a nudge in the right direction. Let $(E,\mathcal{M},\mu)$ be a measure space with $0 < \mu(E) < ...
1
vote
1answer
81 views

Continuous embedding of $L^p$ into $L^q + L^\infty$

Let $L^p$ denote the Lebesgue-space over a $\sigma$-finite measure space $(\Omega,\mu)$. It is known that $L^{p_0} \cap L^{p_1} \hookrightarrow L^p \hookrightarrow L^{p_0} + L^{p_1}$ continuously ...
9
votes
3answers
230 views

Properties of $\bigcap_{p > 1} \ell_p$

Consider the following space of sequences $$\left\{a=(a_n)_{n\in\mathbb{N}}:a\in\bigcap_{p>1}\ell_p, a_n\in\mathbb{R}\right\}$$ What are some of its properties? What is its relation to $\ell_1$ and ...
3
votes
1answer
119 views

$L^\infty$ and the intersection of the spaces $L^p$

I'm trying to understand if it's true that: " if $f\in L^p\quad \forall p\in N\implies f\in L^\infty$"? My thoughts: Since $\int_R |f(x)|^p dx<\infty\quad\forall p\implies |f(x)|\to 0$? Can anyone ...
1
vote
1answer
162 views

Monotonicity of $L^p$ norms

Let $(X, \mathscr{A})$ be a measurable space, and let $\mu$ be a measure on $(X, \mathscr{A})$ such tat $\mu(X) = 1$ suppose that $1\leq p_1 < p_2 < +\infty$. Show that if $f$ belongs to ...
5
votes
4answers
141 views

$\int f_k\to 0 $ but $f_k $ does not converge to $0 $ ae, where $ f_k $ is defined in $[0, 1] $

Give an exemple, in [0, 1], of a sequence of functions $ f_k $ such that $||f_k||_ 1=\int |f|_k \to 0 $ but $ f_k $ does not converge to $0 $ a.e.
2
votes
1answer
67 views

$\int_{B} f_n \phi \rightarrow 0$ if the Weak-$L^p$ norm of $f$ tends to zero?

Let $f_n \in L^p(B)$ be a sequence where $B$ is some ball in $\mathbb{R}^n$. Assume that $\|f_n\|_{L^p(B)} \rightarrow 0$ when $n\rightarrow \infty$, then by some $\phi \in C^\infty_0(B)$ applying ...
1
vote
1answer
265 views

Is a $L^p$ function almost surely bounded a.e.?

I just have a quick question related to $L^p$ spaces. Any help is greatly appreciated. Is it true that if a function $f$ belongs to $L^p$ space, absolute value of $f$ raise to the power of $p$ is ...
1
vote
1answer
219 views

Are all the norms of $L^p$ space equivalent?

Are all the norms of $L^p$ space equivalent? That is, for any $p,~q \in R^+$, there exist two positive number $C_1,~C_2$ such that $$ C_1\|u\|_{L^q} \le \|u\|_{L^p} \le C_2\|u\|_{L^q}. $$
0
votes
0answers
39 views

Lp Space conjugate function [duplicate]

Suppose $1<p,q<\infty$, $\dfrac{1}{p} + \dfrac{1}{q}=1$, $(X,\Sigma,\mu$ is $\sigma$-finite measure space and $g$ is a measurable function such that $fg \in L^1(X)$ for every $f \in L^p(X)$. ...
0
votes
1answer
42 views

Approximation in $L^2$ of functions with values in a convex set

Here is my problem : Let $K$ be a convex set of $\mathbb{R}^m$ ($m\in \mathbb{N}^*$), such that $0$ belongs to the interior of K, I want to approximate (in $L^2(\mathbb{R}^m,\mathbb{R}^m)$) a function ...
1
vote
1answer
58 views

About the convergence of a sequence in $L^1$

Suppose that $f_n$ is a sequence of nonnegative functions such that $\int f_n d\mu=1$ for all $n$, and $f_n\to f$ in $L^1$. Let $p>1$. Is it then true that $f_n^p\to f^p$ in $L^1$?
0
votes
1answer
60 views

If $u\in L^p$, is $u\in L^q$ for some $q>p$?

(Motivation is below) Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Let $p\in [1,\infty)$ and $u\in L^p(\Omega)$. Is there any $q>p$ such that $u\in L^q(\Omega)$? I already know that ...
2
votes
1answer
54 views

Convergence of $\varphi_n(x):=\frac{\varphi(nx)}{n}$ in Schwartz space

I want to find all $\varphi\in\mathcal S(\mathbb R)$ for which the sequence $\varphi_n(x):=\frac{\varphi(nx)}{n}$ converges in $\mathcal S(\mathbb R)$. The first step, I have already managed to do by ...