Questions tagged [lp-spaces]

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 problem is a $L^p$ space.

4
votes
1answer
46 views

Exercise in Holder's inequality

The following is a problem from Royden and Fitzpatrick's Real Analysis book. Find the values of the parameter $\lambda$ for which $$ \lim\limits_{\epsilon\rightarrow0^{+}} \frac{1}{\epsilon^\...
5
votes
1answer
39 views

If $\left(\sqrt{\gamma_n}\right)_{n\in\mathbb N}$ is $L^2(\mu)$-Cauchy, does $\left(\gamma_n\right)_{n\in\mathbb N}$ converge in $L^1(\mu)$?

Let $(\Omega,\mathcal A,\mu)$ be a measurable space Suppose $(g_n)_{n\in\mathbb N}$ is a sequence of bounded $\mathcal A$-measurable functions $g_n:\mathbb R\to\mathbb R$ such that $\left(\sqrt{\...
3
votes
1answer
65 views

Bounded Sequence in $L^\infty$ and Interpolation in $L^p$

a) Let $1\leq p_1\leq p\leq p_2\leq \infty$ and for $\alpha \in [0,1]$ $\frac {1}{p}=\frac {\alpha}{p_1}+\frac {1- \alpha}{p_2}$ Prove that if $f\in L^{p_1}\cap L^{p_2}$, then $f\in L^p$ and we have ...
0
votes
1answer
35 views

Verification of alternative proof of $\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$

I have to show that $$\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$$ Suppose $u\in L^\infty (E) $ for measurable $E \subset \mathbb{R}^d$ having finite measure. I come up with this proof: $$\Big| \|u\|_p -...
1
vote
0answers
35 views

Show {$e^{2\pi i bnx}$} is a basis for $L^2[0,b^{-1}]$

Let $b > 0$ be a fixed positive scalar. Show {$e^{2\pi i bnx}$} for $n \in \mathbb{Z}$ is a orthogonal (but not orthonormal) basis for $L^2[0,b^{-1}]$. I was able to show it is orthogonal and not ...
4
votes
2answers
42 views

I want to prove that two norms are equivalent but I am struggling with an upper bound

I want to compare the usual norm on $L^2(-1,1)$ with the following: $$ \Vert f \Vert_H^{2} = \int_{-1}^1 \vert f(x) \vert^2 \frac{1}{1+x^2}dx $$ Now, for sure I have this:$$\Vert f \Vert_H \leq \...
2
votes
0answers
38 views

Continuity of superposition operator

Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ $$ \varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot)) $$ is known as the ...
1
vote
0answers
33 views

How to show $\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu$

Let $(g_{n})_{n}$ be bounded and measurable where $g_{n}\xrightarrow{n \to \infty} g, \mu-$a.e. and $f_{n} \xrightarrow{L^{p}}f$. I need to show that $g_{n}f_{n} \xrightarrow{L^{1}}gf$, and my proof ...
3
votes
0answers
51 views

Confusion on $\vert\vert \varphi\vert\vert_{p}$ where $p \in [1,\infty[$

Let $\varphi \in C^{\infty}(\mathbb R^n)$ and for $\epsilon > 0$ define $\varphi_{\epsilon}(x):=\epsilon^{-n}\varphi(x/\epsilon)$ such that $\varphi_{\epsilon} \in C^{\infty}(\mathbb R^n)$ with ...
2
votes
1answer
68 views

Need Hint; Show that the Limit Exists when $f\in C^1(0,1)$ and…

The problem is as follows: Assume that $f\in C^1(0,1)$ and $$ \int_{(0,1)}x|f'|^p\,dx<+\infty\qquad\text{for some }p>2. $$ Show that $\lim_{x\rightarrow 0^+}f(x)$ exists. Note: $C^1(0,1)$...
2
votes
1answer
44 views

Upper estimate for an integral in norms

Is there any reference to obtain an upper bound for $$ \int_{\Omega}v_{t}^2(v^2+1/v^2)dx $$ where $v\in H^{2}(\Omega)\cap H_{0}^1(\Omega) \setminus \{0\}$, $v_{t}\in H_{0}^1(\Omega) \setminus \{0\}$ ...
0
votes
0answers
44 views

Hp norm vs Lp norm

I'm reading the wikipedia page on Hardy spaces, and it states that an $H^p$ function can be extended to an $L^p$ function on the boundary of the circle, and in fact $||\tilde{f}||_{L^p} = ||f||_{H^p}$....
0
votes
1answer
53 views

Convergence in $L^p$ but not pointwise a.e.?

I read in the lecture notes of integration that if a sequence of functions $f_n$ converges in $L^p$ to $f$, then there is a subsequence that converges pointwise a. e. to $f$. So does that mean that it ...
0
votes
1answer
29 views

$c_{00}$ is not complete

I try to show that the space $c_{00}=\{(x_n):x_n=0 \text{ all but finitely many }n\}$ is not complete with respect to the norm $\|x\|_\infty=\max |x_n|$. My attempt: Let $(z_n)=\left(1,\frac{1}{2},\...
1
vote
1answer
29 views

Determine $\varphi_{\epsilon}$

Let $\varphi \in C^{\infty}(\mathbb R^n)$ with compact support. For $\epsilon > 0$ define $\varphi_{\epsilon}(x):=\epsilon^{-n}\varphi(x/\epsilon)$ such that $\varphi_{\epsilon} \in C^{\infty}(\...
0
votes
1answer
24 views

Problem on property of sequences of functions in Lp spaces

I was going through the problems from an exam in Measure and Integration Theory and I stumbled upon a problem that stumped me. It is as follows: Let $(X,\mu)$ be a measure space and let $p_1, p_2 \...
3
votes
1answer
42 views

$L^{q,\infty}(\mu) \subset L^p(\mu)$ with $1 \leq p<q <\infty$ and $\mu$ finite measure?

Let $\mu$ be a finite measure on a measurable space $(X,\Sigma)$. I want to prove that exists $C > 0$ so that $$ \| f\|_p \leq C \| f \|_{q,\infty}$$ when $1 \leq p < q < \infty$, where $$ \| ...
3
votes
2answers
52 views

Non compacity of a subset of $\ell^1$

Consider $l^1 = \left\{ \{a_n\}_{n=1}^\infty : a_n \in \mathbb{R}, \ \ \sum_{n=1}^\infty |a_n| < \infty \right\}$ and let $K = \left\{f \in l^1 : |f(k)| \leq \frac{1}{k} \ \ \forall k\right\}$. ...
0
votes
0answers
27 views

Generalisation: isomorphism of sequences spaces

As is well known, the isometric isomorphism $(c_0)^* \cong \ell^1$ holds. Is there an analogous statement for general $L^p$ spaces? Perhaps a good start would be to wonder about generalizations of ...
1
vote
2answers
59 views

$L^1$ convergent subsequence of increasing functions uniformly bounded in $L^2$

Problem: Suppose $f_n \colon [0,1] \to [0,\infty)$ for $n \in \mathbb{N}$ are increasing functions which are uniformly bounded in $L^2([0,1])$. Show that there exists a subsequence which ...
1
vote
2answers
75 views

S is a dense subset of $L^{p'}$, $\int_{E}fg = 0$ for all $f \in S$, then $g= 0$

Problem: $E$ is a measurable set and $1 \leq p < \infty$. Let $p′$ be the conjugate of $p$, and $S$ is a dense subset of $L^{p′}(E)$. Show that if $g \in L^p(E)$ and $\int_{E}fg = 0$ for all $f \...
0
votes
0answers
42 views

Approximation of piecewise linear functions by constant function

Let $f(x) = \begin{cases} A_1 x + B_1 &\mbox{if } x \in [a, x_0] \\ A_2 x + B_2 & \mbox{if } x \in [x_0, b] \end{cases}$ and $f \in C[a, b]$ i.e. f has the "angle" form. Denote $E_t(f) = ||...
1
vote
1answer
37 views

Verification of convergence of random variables

Let $(X_n)_{n \in \mathbb{N}}$ a series of random variables with: $$P(X_n = 2^n) = \frac{1}{2^n} \hspace{15pt}\text{and}\hspace{15pt} P(X_n = 0) = 1-\frac{1}{2^n}$$ for all $n \in \mathbb{N}$. ...
1
vote
1answer
58 views

Do weak convergence and convergence of norms imply convergence in $L^1$?

I know this does hold in $L^2$, since it's a Hilbert space. I suspect that this is not true, but I cannot think of a counterexample. Specifically, I want to know if $f_n \xrightarrow{w} f$ and $\...
4
votes
1answer
127 views

Showing that an integral operator on $L^p$ spaces has a certain norm

Let $X$ be a sigma-finite measure space, and let $k$ be a measurable function on $X\times X$. Suppose that $F(x)=\int |k(x,y)| dy$ and $G(y)=\int |k(x,y)| dx$ are in $L^\infty$. Let $1<p<\...
1
vote
1answer
48 views

Approximating the Lebesgue measure on the full real line with discrete measures

For any $L>0$, let $\mu_L$ be the real Borel measure defined as follows: $$ \mu_L=\frac{1}{L}\sum_{x\in\mathbb Z}\delta_{x/L},$$ $\delta_{x}$ being the atomic (Dirac) measure on $x$, and let $L^p(...
6
votes
1answer
428 views

Show that $e^{X^2/2} \in L^1$ iff $e^{XY} \in L^1$ iff $e^{|XY|} \in L^1$

let $X, Y$ be two identically distributed (both are $\mathcal{N}(0,1)$) independent random variables show that $e^{\frac{X^2}{2}} \in L^1 \iff e^{XY} \in L^1 \iff e^{|XY|} \in L^1$. my attempt : ...
0
votes
1answer
48 views

Determine the LP spaces that contain a given function

I was assigned this excercise: Determine $p$ so that $f(x)=\frac{1}{x^3 - 1}$ belongs in the $\mathfrak{L}^P(E)$ space, where $E=\mathbb{R}-\{1\}$ I've proceeded along this direction: $E$ is ...
1
vote
0answers
29 views

Equiintegrability of some family of sequences

Let $(\rho_n)_n\subset L^1$ be a Dirac sequence. Study the equiintegrability of the following family: (a) $f_n=\rho_n^2$, (b) $g_n=\rho_1*\rho_n$ (convolution), (c) $g_n=\rho_1*\rho_n^2$. My ...
3
votes
1answer
54 views

How is $\|f\|_p \geq \bigg( \int_A |f|^p d \mu \bigg)^{1/p}$?

How is $\|f\|_p \geq \bigg( \int_A |f|^p d \mu \bigg)^{1/p}$? This is presented as "trivial inequality". However since this is essentially the definition of $\|f\|_p$, then I don't see how $>$ is ...
-3
votes
1answer
43 views

Find the value of p for which the following series converges? [closed]

Let $x = (x_1, x_2, \ldots )\in l^4$, $x\neq 0$. For which one of the following values of $p$, the series $$ \sum_{i=1}^{\infty} x_i y_i $$ converges for every $y = (y_1, y_2, \ldots )\in l^p$? A) 1 ...
1
vote
0answers
32 views

Levy upward in $L^2$

Let $(F_n)_n$ be a filtration and $F=\sigma(\cup_n F_n)$. Let $Z$ be a random variable such that $\mathbb{E}Z^2<\infty$. Let $X_n=\mathbb{E}[Z|F_n]$. Given: If $X_n\to Z$ a.s. then $||X_n-Z||_2\...
4
votes
1answer
42 views

Show that $(y_n)\in l^q$

The problem I'm trying to solve is: Let $1\leq p, q\leq\infty$ such that $\frac{1}{q}+\frac{1}{p}=1$. If $$\sum_{n=1}^{\infty} |x_n||y_n|<\infty$$ for all $(x_n)\in l^p$, then $(y_n)\in l^q$. $\...
0
votes
1answer
33 views

Show that a sequence admits converging subsequence

I don't know how to solve the following exercise. I think I should use Ascoli-Arzelà's theorem, but I don't know how. Let $\{ u_n\}_n$ be a sequence of functions in $C^1[0,1]$ with $u_n(0)=0$ for ...
0
votes
1answer
47 views

Showing the Fourier transform of $C_c^{\infty}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$

I am trying to show that $\left\lbrace \widehat{\phi} : \phi \in C_c^{\infty}(\mathbb{R}) \right\rbrace$ is dense in $L^2(\mathbb{R})$, where $\widehat{\phi}$ is the Fourier transform of the function $...
1
vote
2answers
64 views

$L^1(\mathbb R)$ convergence of min and max functions.

Let $f_k , f \in L^1(\mathbb R)$ , with $f_k,f \geq 0$ a.e. in $(0,1)$. Suppose that $f_k\rightarrow f$ pointwise a.e. in $(0,1)$ and that: $\int_0^1 f_n dx \rightarrow \int_0^1 fdx$ Prove that: ...
2
votes
2answers
42 views

What's the meaning of denoting $\ell^p$ as $\ell^p(\mathbb{N})$?

What's the meaning of denoting $\ell^p$ as $\ell^p(\mathbb{N})$? I read that it's like $\ell^p$ "over $\mathbb{N}$". But $l^p$ is sequences indexed by $\mathbb{N}$. So it seems weird to treat the ...
2
votes
1answer
53 views

Continuity of a function between $l^p$ spaces

Let us consider the function defined as $$ F: l^4 \rightarrow l^6 \\ (x_1, \dots,x_n, \dots) \mapsto (x_1^{20}, \dots, x_n^{20}, \dots) $$ I am asked to prove whether this function is continuous or ...
3
votes
1answer
73 views

Convergence in $L^{3/2}$ and in $L^2$

Let $(f_n)_n$ be a bounded sequence in $L^3(\mathbb R)$, such that $f_n\rightarrow f$ in $L^{3/2}(\mathbb R)$. Prove that $f_n\rightarrow f$ converges in $L^2\mathbb (R)$. I have and idea to first ...
2
votes
0answers
82 views

Critique my proof: Proving that $C[a,b]$ is not a Hilbert Space under $L^2$ norm.

So, I would appreciate any critiques I can get for my proof of the following problem. The Problem: Let $[a.b]$ be the closed, bounded interval of real numbers. Show that the $L^2[a,b]$ inner product ...
0
votes
0answers
30 views

Defining an “improper” Lebesgue integral?

I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $\frac{\sin x}{x}$. While I understand perfectly well why such a function ...
0
votes
1answer
51 views

Find the orthogonal complement on $L^2[0,1]$ of all polynomials.

The problem: Determine the orthogonal complement on $L^2[0,1]$ to all polynomials. My approach and intuition thus far: I know for sure intuitively that the orthogonal complement would just be the ...
2
votes
1answer
42 views

Functional Space Inequality for Sobolev Space and Lp Space

Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
1
vote
1answer
38 views

Why is the conditional expectation of an $L^{p}$-function again in $L^{p}$?

Let $(\Omega, \mathcal{A},P)$ be a probability space and let $X,Y\colon \Omega \rightarrow \mathbb{R}$ be random variables. Furthermore, let $Y$ be $p$-integrable. Then why is the conditional ...
0
votes
0answers
46 views

Proof of infinity matrix norm

Given the $l_{\infty}$ matrix norm for $A{\in}{\Bbb{R}}^{mxn}$ is defined as: $\|A\|_{\infty} =\max_{1 \leq i \leq n}\|a^{i}\|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A), Show that: $\|A\|...
3
votes
2answers
55 views

Does $L_1$ convergence of continuous functions imply pointwise convergence?

Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $x\in[0,1]$? I'm pretty sure the answer is no, ...
3
votes
1answer
87 views

Representation of linear operator between $L^p$ spaces.

I was wondering where I could find a reference to the a characterization of continuous linear operators: $$T:L^p(X,\mu)\to L^q(Y,\eta)$$ of the form $T(f)(y)=\int_{X} k(x,y)f(x)d\mu$ for some $k$ ...
3
votes
1answer
39 views

Proving an inequality of random variables

I am currently reading a paper that claims the following fact: Let $X$, $Y$ be $L^2$ random variables on some probability space. The $L^2$ norm is denoted by $\| \cdot \|_{2}$. Then there exists $C&...
1
vote
0answers
82 views

Show that $(f_n)_n$ is relatively compact in $L^p$ space

Let $I=[0,1]$, $Q=I\times I$ and $(u_n),(v_n)$ bounded sequences in $L^2(I)$. Assume $x\mapsto u_n(x), x\mapsto v(x)$ are continuous and monotone non decreasing on $I$ for all $n\in\mathbb{N}$; define\...
1
vote
1answer
47 views

Have I understood Compact Set correctly

In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that $\overline{\{x \in \...