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 ...

learn more… | top users | synonyms

1
vote
1answer
23 views

If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?

This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it. Consider a probability measure space $(X,\Sigma,\mu)$ and ...
2
votes
1answer
126 views

Continuous function bounded in $L^\infty$

Is a continuous (real-valued) function in $L^\infty$ a (everywhere-)bounded function?
1
vote
1answer
43 views

$L^p$ convergence proof check

I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it. Theorem: Let $\{u_i\}$ ...
1
vote
1answer
65 views

Showing that one cannot continuously embed $\ell^\infty$ in $\ell^1$.

Is it possible to embed $\ell^\infty$ into $\ell^1$ continuously? I.e. can one find a continuous linear injection $I:\ell^\infty \to \ell^1$. I have reduced a problem I have been working on to ...
3
votes
1answer
158 views

Convergence of $L^p$ norm as $p \downarrow 0$ [duplicate]

Consider a measurable space $(\Omega, \mathscr{F}, P)$ with $P(\Omega) = 1$. Define for measurable functions $X$ the following $\| X \|_p := \left(\int |X|^p dP\right)^{1/p}$. We know that for $p \in ...
1
vote
1answer
25 views

Remainder of a series converges uniformly?

Let $B \subset \Bbb R^{\Bbb N}$ and $p \geq 1$. Suppose $$ \sup_{u\in B}\sum_{n=0}^\infty |u_n|^p \leq 1,\qquad \sup_{u\in B}\sum_{n=0}^\infty |u_{n+1}-u_n|^p\leq 1 $$ Is it true that $$ \sup_{u\in ...
2
votes
2answers
140 views

Compact inclusion in $L^p$

Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$? What is the counterexample if what I said is wrong? Thank you.
5
votes
1answer
144 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
1
vote
2answers
121 views

Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?

Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$. Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
1
vote
1answer
42 views

Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property

$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $. To prove ...
8
votes
2answers
118 views

Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by ...
12
votes
1answer
125 views

Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
3
votes
1answer
61 views

Monotonicity of $\mathcal{l^p}$ spaces using only Hoelder inequality

For $p > 0$, let $\ell^p$ be the space of sequences for which  $$\sum_{i=1}^{\infty} |a_i|^p$$  is finite ($a_i \in \mathbb{R}$). It is well-known that, for $q > p$, $$\ell^p \subset \ell^q.$$ ...
2
votes
1answer
101 views

Lebesgue space - $L^p$ spaces

Let $X=\{a,b\}$,and $\mu(\{a\} )=1$, and $\mu(\{b\} )=\mu(X)=+\infty$ and $\mu(\emptyset)=0$. Is it truth that $L^\infty(\mu)$ is the dual space of $L^1(\mu)$. Whether $L^\infty(\mu)=L^1(\mu)^\ast$? ...
1
vote
1answer
153 views

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$. My friends and I have literally been pouring over this problem for days now without success. We've been using Hölder's ...
2
votes
1answer
261 views

Examples of $f \in L^p$ iff $p_0 < p < p_1$, $p_0 \le p \le p_1$ or $p = p_0$

Hi how to show the following: Suppose $0 < p_0 < p_1 \leq \infty$. Find examples of functions $f$ on $(0,\infty)$ with Lebesgue measure such that $f \in L^p$ if and only if (a) $p_0 < p ...
1
vote
0answers
81 views

Inclusion maps on $L^p$

How to show for $1 \leq p < q < r$ the inclusion maps $$L^p \cap L^r \rightarrow L^q$$ $$L^q \rightarrow L^p + L^r$$ are continuous. where the norms are defined in the following: $L^p$ ...
1
vote
1answer
125 views

$\mathcal{L}^p$ spaces and convolution

Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that (a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous ...
1
vote
1answer
32 views

Measure and $L_\infty$ space

Consider a function $f \in L_\infty$. I am trying to see if the following statement is true and if so why. $$ \mu\{\, \vert f \vert = \Vert f \Vert_\infty \} \stackrel{??}{>} 0 \text { and } \{\, ...
1
vote
1answer
103 views

$L^p$ space, paralellogram law

Let $(X,\mathcal{M},\mu)$ be a measure space and $1\leq p\leq\infty$. Suppose $f,g\in L^p(X,\mathcal{M},\mu)$ with $f(x)g(x)=0$ for almost all $x\in X$ and $\Vert f\Vert_p=\Vert g \Vert_p=1$. How to ...
1
vote
1answer
84 views

$l^{p}$ is not finite dimensional

Well, the exercise was to prove that $l^{p}$ is not finite dimensional space for $p$=2. I did it proving that the unit ball is not compact. Easy. However, i was trying to build an element $x \in ...
3
votes
1answer
125 views

$W^{1,p}$ compact in $L^\infty$?

Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
3
votes
2answers
194 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
2
votes
2answers
104 views

$L_p$ space,convergence

Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to ...
3
votes
0answers
149 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
3
votes
2answers
177 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
1
vote
1answer
1k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms.

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
3
votes
2answers
61 views

Proving that certain subspace of $\ell_1$ is non closed

I need to prove that $$L= \left\{(x_i) \in\ell_1 : \sum_{i=1}^\infty ix_i= 0\right\}$$ is non-closed in $\ell_1$. I can't really think of sequences of sequences that are in this subspace, much less ...
6
votes
1answer
76 views

$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$
3
votes
1answer
99 views

Are Hoelder spaces continuously embedded into Lp spaces?

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain of class $C^1$ and $\gamma\in (0,1]$. I am only interested in Hoelder spaces of form $C^{0,\gamma}(\overline{\Omega})$, i.e. I am not interested in ...
0
votes
1answer
109 views

When is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$? [duplicate]

Given a measure space $(X,\Sigma,\mu)$, when is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$ for $p>r$, or for $p<r$ . Thanks.
3
votes
2answers
99 views

Show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ is not closed in $\ell^2$

How to show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ where $e_n=(\delta_{nk})_{k\in\mathbb N}$ is not closed in $\ell^2$?
1
vote
1answer
75 views

Inclusion in $L_p$ space

I have been wondering how to prove the following statement, and would greatly appreciate your help: If $f$ is a bounded function on $E$ that belongs to $L_{p_1}(E)$, then it belongs to $L_{p_2}(E)$ ...
2
votes
1answer
418 views

Norm equivalence Sobolev space

I have this problem: Let $k>0$ (integer) and $1 \leq p < \infty$. Show that the norms $$ ||u||_{W^{k,p}(U)} = \bigg( \sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^{p}(U)}^{p}\bigg)^{\frac{1}{p}} $$ ...
0
votes
2answers
118 views

How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
2
votes
2answers
106 views

Does $e_n(x)=\exp\left( \frac{i \pi n}{N}x \right)$ define an orthonormal basis of $L^2(-N,N)$?

We know that the Fourier system is complete, i.e. that $\lbrace e_n: ~ n \in \mathbb{N} \rbrace$ defined by \begin{equation} e_n(x)=\frac{1}{\sqrt{2 \pi}}\exp(inx), ~~~ n \in \mathbb{Z} \end{equation} ...
2
votes
2answers
148 views

Weak limit of disjoint normalized sequence in $L^p$

I want to prove that the weak limit of a disjoint normalized (pairwise disjoint supports, elements of norm $1$) sequence $(f_n)$ in $L^p$ for $p >1$ is zero ? I started with the measure of ...
0
votes
1answer
80 views

Is an orthnormal basis of $L^2([0,1])$ also an orthonormal basis of $L^2((0,1))$?

My question is: If $\lbrace e_n \rbrace$ is an orthnormal basis of $L^2([0,1])$, is $\lbrace {e_n}_{|(0,1)} \rbrace$ an orthonormal basis of $L^2((0,1))$? As the points $\lbrace 1 \rbrace$ and ...
1
vote
0answers
440 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
2
votes
1answer
130 views

Uniform convergence in $L^p$-spaces

Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$. ...
3
votes
3answers
103 views

Is the inclusion map of $\ell^1$ into $\ell^2$ a closed map?

In particular, I'm interested in the subset $\{x : \lVert x\rVert_1 \ge 1\}$ inside $\ell^2$. Is this a closed subset? thank you!
2
votes
1answer
91 views

Question on $L^p$ spaces

Let $f:[0,1] \to \mathbb{R}$ be a measurable function. Prove that $$ f \in L^\infty([0,1];\mathbb{R}) \iff f \in L^p([0,1];\mathbb{R}) \ \ \forall p \ge 1 \ \text{ and } \sup_{p\ge ...
3
votes
1answer
50 views

Why $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$?

How show that $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$? Can someone help me? Thank you!
2
votes
1answer
187 views

About a property of the Dirac delta function

How can I show that there is no $u$ satisfying both (i) and (ii):$$(i) \; u \in L^p (\Bbb R^n )$$ and $$(ii) \int_{\Bbb R^n} \delta (x) \phi(x) dx= \int_{\Bbb R^n} u (x) \phi(x)dx\; ( \forall \phi ...
2
votes
1answer
100 views

How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?

I not solve the follow limit $$\lim_{p\rightarrow 0} \bigg[\int_{\Omega} |f|^p d\mu \bigg]^{1/p} = \exp\bigg[ \int_{\Omega} \log|f|d\mu \bigg],$$ where $(\Omega, \mathcal{F}, \mu)$ is a probability ...
1
vote
1answer
44 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
0
votes
1answer
178 views

Essential Supremum

For $f\in L^\infty[a,b]$, show that $$\|f\|_\infty = \min \big\{M : m\{x \in [a, b] : |f(x)|>M\} = 0\big\}\;,$$ and if, furthermore, $f$ is continuous on $[a, b]$, that $\|f\|_\infty = ...
6
votes
1answer
284 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
8
votes
1answer
330 views

Distance minimizers in $L^1$ and $L^{\infty}$

If $H$ is a Hilbert space, we have the Hilbert Projection Theorem, which tells us that given a nonempty, closed, convex subset $K \subset H$, and a point $x \in H$, there is a unique point $y \in K$ ...
3
votes
1answer
89 views

Function in $L^1([0,1])$ that is not locally in any $L^{\infty}$

Can we find a function such that $f\in L^1([0,1])$ and for any $0\leq a<b\leq 1$ we have that $||f||_{L^{\infty}([a,b])}=\infty$?