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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2
votes
2answers
32 views

Show that a subspace is proper dense in $l^1$ sequence space. L^1 space.

Let Y = $L^1 $($\mu$) where $\mu$ is counting measure on N. Let X = {$f$ $\in$ Y : $\sum_{n=1}^{\infty}$ n|$f(n)$|<$\infty$}, equipped with the $L^1$ norm. Show that X is a proper dense subspace ...
2
votes
0answers
113 views

Characterization of $L^p$ function for $p>1$

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, $1<p<\infty, 1/p +1/q=1,f \in L^1(\Omega)$. Is it true that if there exists $C>0$ such that $sup \{\int fg: g$ is a step function, $ ...
2
votes
2answers
47 views

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?
2
votes
1answer
47 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
0
votes
1answer
14 views

Does $\partial_b u(\cdot,b) \in L^2(A)$ for fixed $b$ imply that $u(\cdot,b) \in L^2(A)$?

Let $u$ be defined on $A \times B$ where $A$ and $B$ are two bounded domains, and write $u=u(a,b)$. Suppose that the weak derivative $\partial_b u(\cdot,b) \in L^2(A)$ for fixed $b$. Does this imply ...
4
votes
1answer
39 views

$\||f|^{2}f\|_{H^{s}}\leq C \|f\|^{2}_{L^{\infty}} \|f\|_{H^{s}}$ for $s>0, f\in H^{s}(\mathbb R)$?

We let $H^{s}(\mathbb R^{n}), (s\in \mathbb R)$ the Sobolev spaces. It is well known that: the space $H^{s}(\mathbb R^{n})$ is an algebra with respect to pointwise multiplications, for $s>n/2.$ ...
2
votes
1answer
66 views

Prove that $T$ is bounded if $\langle Tx, y \rangle = \langle x, T^*y \rangle$

Suppose $T: L^2(\mathbb R^d) \to L^2(\mathbb R^d)$ is a linear operator, and there exists $T^*: L^2(\mathbb R^d) \to L^2(\mathbb R^d)$ such that $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for ...
2
votes
3answers
51 views

$\|fg\|_{L^2(\Omega)}\leq \|f\|_{L^2(\Omega)}\|g\|_{L^\infty(\Omega)}$

is true that $\|fg\|_{L^2(\Omega)}\leq \|f\|_{L^2(\Omega)}\|g\|_{L^{\color{blue}\infty}(\Omega)}$ ? I can't see a proof for this :/ ( of course, $\|fg\|_{L^2(\Omega)}\leq ...
2
votes
1answer
45 views

Find all functions such that $ (\int _0 ^1 xf(x) dx)^3 = \frac{4}{25} \int _0 ^1 f(x)^3 dx$

Calculate all the functions $f \in L^3$ such that $$ \left(\int _0 ^1 xf(x) dx\right)^3 = \frac{4}{25} \int _0 ^1 f(x)^3 dx$$ Can someone please walk me through this because there are no such ...
2
votes
1answer
23 views

Convergence of a product of sequences convergent in mean when one of them is bounded

Suppose $X_n\to X$ in $L^1$ and $V_n\to V$ in $L^1$ and $(V_n)$ is a bounded sequence. I'm trying to show that then $\mathbb{E}X_nV_n\to \mathbb{E}XV$. One has for all $N\in\mathbb{N}$ ...
0
votes
2answers
40 views

$\int_{\mathbb R}|f(x)|^{2} dx <\infty \implies \sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx <\infty$?

Let $f\in L^{2}(\mathbb R),$ that is, $\int_{\mathbb R}|f(x)|^{2} dx <\infty,$ and $\beta>0.$ My Question: Is it true that that: $\sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx ...
3
votes
1answer
42 views

Convergence in $L^1_{loc}$ implies convergence almost everywhere

Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ ...
1
vote
1answer
42 views

is $L^2 (\mathbb R)\subset L^\infty(\mathbb R)$?

I know that because i'm working on an infinite-measure space it could be tricky. And from my experience the answer to my question is probably no.. But nevertheless, I can't think of a non-bounded ...
0
votes
1answer
9 views

Domain of a linear operator and using Interpolation Theorems

This question may be very elementary but I want to make sure. Many of the interpolation theorems require a linear operator to be defined on the sum of two spaces, for example $T$ is a linear ...
2
votes
0answers
31 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
1
vote
0answers
21 views

Dual of $L^\infty(I,H^1(M))$

What is the dual of $L^\infty(I,H^1(M))$? Any references? Where $H^1(M)$ is Sobolev space, and $I$ is some interval in $\mathbb{R}$, and $M$ is a compact manifold, like the $n$-dimensional torus.
2
votes
1answer
32 views

Prove one limit

The task is to show that if $f \in L^{\infty}\left(\mathbb{R}\right)$, then $$\lim\limits_{n\to ...
2
votes
1answer
27 views

Does $f _n \to f $ pointwise imply $f _n $ converges to $f $ in $L ^p $ norm if $\{f_n\}$ is Cauchy in $L^p$?

If a sequence $\{f _n \}$ of functions converges pointwise to a function $f $ does this imply that if the same sequence is a Cauchy sequence in some $L^p $ norm then it converges to the same function ...
3
votes
1answer
53 views

Prove that $\{\sin x, \sin 2x, … , \sin nx\}$ is a linearly independent set

Prove that $\{\sin x, \sin 2x, ... , \sin nx\}$ is linearly independent. The short solution that I do not understand is as follow: For p and q are positive integer, we have $$ ...
1
vote
1answer
15 views

Condition on $f$ in $L^{p, \infty} $ implies $f \in L^q$

If $f \in L^{p, \infty}$ and $\mu ( \{ x : f(x) \neq 0 \} ) < \infty$, then $f \in L^q$ where $q <p$. $\textbf{My Attempt:}$ Let $E = \{x:f(x) \neq 0 \}$ $$ || f ||_q^q = q \int_{0}^{\infty} ...
0
votes
0answers
17 views

Show f is in Lp if and only if sum of measures s.t. |f(x)|>2^n converges [duplicate]

Let $p \in [1,\infty)$ and suppose $\mu$ is a finite measure. Prove that $f \in L^p(\mu)$ if and only if $$\sum^\infty_{n=1} (2^n)^p \mu(\{x:|f(x)| > 2^n\}) < \infty$$ I spent a while trying to ...
2
votes
1answer
27 views

Lebesgue's differentiation theorem for all points

Let $f \in L^2(0,T)$ be such that $f(t)$ is well-defined for every $t$ (not just a.e. $t$). But I have no continuity of $f$. We have by Lebesgue's differentiation theorem that $$\lim_{a \to ...
2
votes
1answer
17 views

If $u_n \to u$ in $L^2(\Omega)$, and $u_n \in L^\infty(\Omega)$, is $u \in L^\infty(\Omega)$?

If $u_n \to u$ in $L^2(\Omega)$, and each $u_n \in L^\infty(\Omega)$, is $u \in L^\infty(\Omega)$ too? There is no uniform bound on $u_n$ in $L^\infty$ though. I don't think it is.
1
vote
1answer
10 views

Linear functionals over a non dense subset in $\ell^2$

I have come across the following statement a couple of times, but cannot figure out quite how to justify it: If $A$ is not a dense subset of $\ell^2$ then its closure is not all of $\ell^2$ so ...
0
votes
1answer
29 views

$\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$?

Let $u\in H^{1}(\mathbb R).$ Is Gagliardo–Nirenberg interpolation inequality valid for the $p=3, q=r=2, m=1, 0< \alpha < 1$ ; and $j=0$ ? That is, is it true that,$\|u\|_{L^{3}(\mathbb R)} ...
0
votes
1answer
47 views

Let $M=\{f(x) \in C[0,1]\mid f(0)=0\}$. Is $\overline{M}=L^2[0,1]$?

Let $M=\{f(x) \in C[0,1]\mid f(0)=0\}$. Is $\overline {M}=L^2[0,1]$? The closure is taken under the usual norm of $L^2[0,1]$. Moreover,fix n different points$(a_k)\subseteq[0,1]$,k=1...n, ...
0
votes
1answer
29 views

Calculating the norm of a bounded linear functional

Let $C[0,1]$ be the space of all continuous functions on $[0,1]$. Then for any $g\in C[0,1]$ define the norm on $C[0,1]$ as $\|g\|=\max_{t\in[0,1]}|g(t)|$. Then let $f:C[0,1]\to\mathbb{R}$ as $$ ...
2
votes
1answer
32 views

Convolution with Gaussian, without dstributioni theory, part 3

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
1
vote
1answer
27 views

Convolution with Gaussian, without distribution theory, part 2

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
0
votes
2answers
68 views

Is it true? If $\lim{\int{|f_n-f|^2}} \rightarrow{0}$, then $ \lim {\int{|f_n-f|}} \rightarrow{0}$ (on $ [0,1]$). [closed]

Let $ f_n $ and $f$ be Lebesgue integrable, $ f_n , f \geq 0 $. Is the following statement true? If $ \lim{\int{|f_n-f|^2}} \rightarrow{0}$, then $ \lim {\int{|f_n-f|}} \rightarrow{0}$ (on $ ...
2
votes
2answers
66 views

$\|f\|_{L^{3}(\mathbb R)}^{4} \leq C \|f\|_{L^{2}(\mathbb R)}^{3} \|\nabla f\|_{L^{2}(\mathbb R)}$ ; for some constant $C$?

Let $f\in H^{1}(\mathbb R).$ (Sobolev space) My Question: Is it true that: $\|f\|_{L^{3}(\mathbb R)}^{4} \leq C \|f\|_{L^{2}(\mathbb R)}^{3} \|\nabla f\|_{L^{2}(\mathbb R)}$ ; for some constant ...
2
votes
1answer
57 views

$\|f\|_{L^{3}(\mathbb R)}^{3} \leq C \|f\|_{L^{2}(\mathbb R)} \|\nabla f\|_{L^{2}(\mathbb R)}^{3}$ ; for some constant $C$?

Let $f\in H^{1}(\mathbb R)$ (Sobolev space). My Question: Is it true that: $\|f\|_{L^{3}(\mathbb R)}^{3} \leq C \|f\|_{L^{2}(\mathbb R)} \|\nabla f\|_{L^{2}(\mathbb R)}^{3}$ ; for some constant ...
5
votes
2answers
122 views

Convolution with Gaussian, without distribution theory, part 1

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
5
votes
1answer
120 views

Showing this limit as $x\to 0$

I'm very close to the result and must be missing something basic: $f$ is absolutely continuous on $[\epsilon,1]$ for each $\epsilon \in (0,1)$, $1<p<2$, and, for $0\leq x\leq y\leq 1$, we have ...
1
vote
1answer
40 views

suppose $f:[1,\infty] \rightarrow R, f(1)=0, f'$ is continuous and bounded and $f' \in L^2([1, \infty])$, show $g=f(x)/x \in L^2([1, \infty])$ .

It's a problem in Bass's real analysis book. suppose $f:[1,\infty] \rightarrow R, f(1)=0, f'exist$ and is continuous and bounded and $f' \in L^2([1, \infty])$, let $g(x)=f(x)/x,$ show $g \in L^2([1, ...
3
votes
2answers
39 views

Suppose $\mu(X)=1$ and $f, g $are nonnegative function such that $fg \geq 1$ a.e. prove $(\int fd\mu)(\int gd\mu) \geq 1$.

I have a problem in $L^p$ space. Suppose $\mu(X)=1$ and $f, g $are nonnegative function such that $fg \geq 1$ a.e. prove $(\int fd\mu)(\int gd\mu) \geq 1$. I have no any idea to prove that. Holder ...
1
vote
0answers
20 views

Point wise convergent [duplicate]

Suppose that $f\in L^1(\mathbb{R})$ and that $f(x)=0$ if $x\notin[0,1]$. For $n=1,2,\ldots,$ let $f_n(x)=f\left(x+\frac{1}{n}\right)$. Prove that $\|f_n-f\|_1\to 0$ as $n\to\infty$.
1
vote
1answer
37 views

Why does this set have finite measure?

Fix a measurable $g$ and let $f$ be any simple measurable function that vanishes outside a set of finite measure and satisfies $||f||_p=1$ where $p>1$. If $|\int fg|<M$ for all such $f$, why ...
4
votes
1answer
77 views

Weak convergence in $L^q$ but not in $L^p$, where $1\leq p<q<\infty$?

I've been trying to find a sequence $\{f_n\}$ of functions on $[0,1]$ that converges weakly in $L^q$ but does not converge weakly in $L^p$, where $1\leq p<q<\infty$. I'm stuck and any hints ...
0
votes
1answer
61 views

If $f_n\to f$ a.e. and are bounded in $L^p$ norm, then $\int f_n g\to \int fg$ for any $g\in L^q$

Suppose $p>1$ and $q$ is its conjugate exponent. Suppose $f_n \rightarrow f~a.e.$ and $\sup_n\|f_n\|_p < \infty $. prove that if $g \in L^q,$ then $\lim_{n \rightarrow \infty} \int f_ng=\int ...
2
votes
1answer
54 views

Behavior at $0$ of a function that is absolutely continuous on $[\epsilon, 1]$

The function $f$ on $[0,1]$ is absolutely continuous on $[\epsilon,1]$ for $0<\epsilon<1.$ I further have that $$\int_0^1x|f'(x)|^pdx<\infty.$$ I'm trying to show that $$ \lim_{x\to 0}f(x)\ ...
4
votes
2answers
92 views

Do $L^p$ spaces have the approximation property?

A Banach space $X$ has the approximation property if every compact operator $T:X \to X$ is the norm-limit of a sequence of finite-rank operators. My question is if there is a simple proof that the ...
1
vote
1answer
40 views

How do I prove the interior of subspace $\ell^1$?

Let $E:=\ell^1$ is Banach space with standard norm for $\ell^1$, $P:=\{\bar{x}\in\ell^1: \bar{x}=(x_i)=(x_1,x_2,\ldots),x_i \geq 0, \forall i \in \mathbb{N}\}$ and defined that interior of $P$ is ...
1
vote
1answer
30 views

If $f$ and $g$ have the same $L^{2}$ norm, does it imply $hf$ and $hg$ have the smae $L^{2}$ norm?

Let $f,g \in L^{2}(\mathbb R)$ with $\|f\|_{L^{2}}=\|g\|_{L^{2}}$( that is, $f$ and $g$ have the same $L^{2}-$ norm). We choose $h\in \mathcal{S}(\mathbb R)$(= Schwartz space) so that $hf, hg \in ...
4
votes
1answer
43 views

Convergence of $\operatorname E|X_n|^p$ when $0<p<1$

Let $0<p<1$ and $X_1,\ldots,X_n$ be random variables with finite absolute moments of order $p$. Suppose that the random variables $X_1,\ldots,X_n$ converge in mean of order $p$ to a random ...
2
votes
1answer
31 views

Functions such that $\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$ and …

Can we find a bounded function $a:\mathbb{R}\to\mathbb{R}$ and a function $f\in L^1(\mathbb{R})$ with $f\neq 0$ such that $$\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$$ and ...
2
votes
1answer
38 views

Proof that v belongs to l_p space under certain conditions

I am struggling with the following problem: Let $M >1$ and $\lambda \in (0,1)$, $\mathbf{z} \in \mathcal{l}_p$. If $|v_t|^p < (M\sum_{s=t}^\infty \lambda^{s-t+1} |z_s| )^p $, then $\mathbf{v} ...
1
vote
2answers
28 views

Continuity of integration in $L^1(X)$.

Let $\varepsilon>0$ and let $f\in L^1(X)$. Show that there exists $\delta>0$ such that for any $E\subseteq X$ with $m(E)<\delta$ we have $$ \int_E \lvert f\rvert<\varepsilon. $$ MY ...
5
votes
1answer
75 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
1
vote
0answers
37 views

Approximation of a smooth function by smooth functions with compact support

Let $U \subset \mathbb{R}^N$ be a bounded open set, and let $h \in C^\infty(U)$. Since $h \in L^p(U)$ for every $1 \leq p \leq \infty$, there exists a sequence $(h_j) \subset C_c^\infty(U)$ such that ...