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

$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$

Let $A \neq \varnothing$ and $\cal {F}$$(A) = \{F \subset A \mid F$ is finite$\}$. Define $\ell_2 (A) =L^2(A, 2^A, \mu_C)$, with $\mu_C$ the counting measure. Let $C(A) = \{f: A \to \Bbb C, \exists ...
3
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1answer
27 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
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0answers
30 views

If $f_n$ is in $L^1$ and $f_n$ converges to $f$, is $f$ in $L^1$?

This question might come off as basic to most of you, but this isn't basic to me. If $f_n$ is in $L^1$ and $f_n$ converges to $f$ in $L^1$ ($||f_n-f||_1 \to 0$ as $n\to \infty$), does it necessarily ...
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2answers
48 views

Real Analysis, Folland Problem 6.1.12 $L^p$ spaces

Problem 6.1.12 - If $p\neq 2$, the $L^p$ norm does not arise from an inner product on $L^p$, except in trivial cases when $\dim(L^p) \leq 1$. (Show that the parallelogram law fails.) Attempted ...
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0answers
20 views

Real Analysis, Folland problem 6.1.7 $L^p$ spaces

problem 6.1.7 - If $f\in L^p\cap L^\infty$ for some $p < \infty$, so that $f\in L^q$ for all $q > p$, then $\|f\|_{\infty} = \lim\limits_{q\rightarrow \infty}\|f\|_{q}$. Attempted proof - ...
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2answers
25 views

Is $\ell_q\subset\ell_p$? What's the “biggest” and “smallest” sequence space?

(By smaller and larger, I mean as subsets/supersets of complex sequence spaces.) I have a couple of questions about sequence spaces (over $\Bbb{C}$). I was proving that $(e^{(m)})$ (the constant $0$ ...
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1answer
19 views

Showing an inequality in the proof of $L^p$ is a Banach space for $1\le p \lt \infty$.

This is part of the proof that $L^p$ is a Banach space from Folland's Real Analysis, but there is a part that I don't understand. Suppose $\{f_k\}\subset L^p$ and let $G_n=\sum_1^n |f_k|$ and ...
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1answer
31 views

What are the differences between $l^p$ space and $L^p$ space?

I always thought small $l^p$ space, or so-called Banach space, is the space equipped with a norm similar to vector norm as $$||x||_p = \left( \sum_{i\in\mathbb{N}} |x_i|^p \right)^{\frac{1}{p}}$$ ...
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0answers
9 views

Diagonal argument involving relative compactness of densities

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], ...
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1answer
95 views

Real Analysis, Folland Problem 6.1.2 $L^p$ spaces

Background Information: In this chapter we work on a fixed measure space $(X,M,\mu)$. If $f$ is measurable on $X$ and $0 < p < \infty$, we define $$\|f\|_{L^p} = \left[\int |f|^p ...
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2answers
39 views

Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$

Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) . ...
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0answers
9 views

What is the Stein Interpolation Theorem for any strip?

Stein Interpolation Theorem: Assume $T_z$ is an operator depending analytically on $z$ in the strip $0\leq Re (z) \leq 1.$ Suppose $T_z$ is bounded from $L^{p_0}$ to $L^{r_0}$ when $Re (z)= 0,$ and ...
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0answers
25 views

Sobolev embedding fails for $p=n$

As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously ...
2
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2answers
38 views

Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
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0answers
12 views

Iterated convolutions w.r.t. different variables of a function

I do not understand a claim from a paper: Let $b:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bounded function and let $$b^{n} (t,x) = b(t,x) \ast \psi_n(t) \ast\phi_n(x), $$ where ...
3
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1answer
33 views

Show that $L^1$ function is $0$ a. e.

Suppose $f \in L_1(\mathcal{R})$ satisfies for every measurable $A \subset \mathcal{R}$ $$ |\int_A f| \leq [m(A)]^{(1+\epsilon)} $$ for some $\epsilon >0$. Prove $f=0$ a.e. This is a problem on ...
4
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1answer
54 views

$p$-norm inequality

Let $p \ge 2$ and $q$ such that$${1\over p} + {1\over q} = 1.$$Is it true that there exists a constant $c$ such that for all $x$, $y$ such that $\|x\|_q \le 1$ and $\|y\|_q \ge 1$$$\left\|x - ...
1
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1answer
41 views

What does it mean “a Lebesgue point of $f$”?

Follan - Real Analysis p.243 Theorem 8.15 As far as I know, the term, Lebesgue point $x$, is defined for $L^1$ functions such that $\lim_{r\to 0} \frac{1}{m(B(x,r))} \int_{B(x,r)} |f - f(x)| dm =0$. ...
1
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1answer
36 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
4
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2answers
43 views

Is $f(x)\exp(-x^2)$ summable if $f$ is square summable?

Suppose that $f \in L^2(\mathbb R)$; i.e. $$ \int_{- \infty}^\infty \vert f(x) \vert^2 dx < \infty. $$ Can we from this infer that $$ \int_{- \infty}^\infty \vert f(x)\vert e^{-x^2} \, dx < ...
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1answer
35 views

Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
0
votes
1answer
25 views

subspace of $L^2$

Let $(X,B(X),\mu)$ a measurable space, for a positive finite measure $\mu$, we consider $H=L^2(X,d\mu)$, Let $A$ a closed subspace of $H$, we know that $A$ is a hilbert space, can we say that it exist ...
2
votes
2answers
40 views

If $f \in L^2[-\pi, \pi]$, then $f \in L^1[-\pi, \pi]$ and$\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$

I'm learning about Fourier analysis and need help with the following problem (which is part of a subchapter on $L^p$ spaces): Using the Cauchy-Schwarz inequality show that if $f \in L^2[-\pi, ...
0
votes
1answer
38 views

The Lebesgue Fundamental Theorem of Calculus

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two Lebesgue integrable functions. If we have $$f(b)-f(a)=\int_a^bg(x)dx$$ for almost all $a,b\in \mathbb{R}$. How can we modify $f$ on a set of measure zero to ...
1
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1answer
48 views

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

I have some problems with my notes: my teacher wrote that if a sequence $\{g_j\}_j\subseteq L^{1,\infty}(\Bbb R^n)$ (which is the weak $L^1$ space, endowed with the quasinorm $||\cdot||_{1,\infty}$) ...
2
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0answers
36 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists ...
0
votes
1answer
18 views

Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
0
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1answer
26 views

$L^{\infty}$ convergence for random variable

I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
1
vote
1answer
49 views

If an $H^1$ function vanishes on a set of positive measure, its $L^2$ norm is controlled by the gradient

I am trying to solve question 15 from Evans' PDE book, chapter 5. You have a set of positive measure, subset of the unit ball $B$, such that $u$ is equal to zero on that set. Then, one can show that: ...
2
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0answers
76 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

Let $\Omega\subset \mathbb R^d$ ($d=2,3$) is a bounded Lipschitz domain. Question: Is it true that for each function $g(x)\in L^2(\Omega)$ one can find a sequence $\{g_n\}_1^\infty$ of ...
2
votes
2answers
45 views

Showing that a subspace of $L^p$ is closed

I would like to prove that a particular subspace of $L^p([1,\infty[)$ (for some $p \in [1,\infty[$) is closed, but I'm not sure how to do it properly. For any sequence $(x_n) \in \ell^p$, let ...
2
votes
1answer
50 views

$X$ be Banach , $T:X \to \mathcal l ^{\infty}$ be linear , $(Tx)_n$ the $n$-th term of $T(x)$;$f_n(x)=(Tx)_n$ ; if each $f_n$ is bdd then so is $T$?

Let $X$ be a Banach space , $T:X \to \mathcal l ^{\infty}$ be a linear transformation , for each $x\in X$ and each $n \in \mathbb N$ , $(Tx)_n$ be the $n$-th term of $T(x)$ and for each $n \in ...
1
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1answer
16 views

$f(t)=t^{2r-2p-1} (e^{2t}-e^{-2t})^{-1/2}\in L^{1}(0, \infty)$?

Let $r>3/4,$ and $p>1/2.$ My Question: Can we expect $\int_0^{\infty} t^{2r-2p-1} (e^{2t}-e^{-2t})^{-1/2} dt < \infty$? I am trying to analyze the above integral. Any ...
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0answers
24 views

If $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ uniformly over $\Omega$

Consider the following statement: If $(\rho_{\epsilon})_{\epsilon > 0}$ is an approximation identity and $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ ...
1
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2answers
41 views

Characterization of weak convergence in Lp

Weak convergence in $X=L^p(0,1)$ for $1<p<\infty$ can be characterized as following: $f_n\rightharpoonup f$ if and only if $f_n$ is bounded in $X$ and $\int_{(0,t)}\;f_n\;\rightarrow ...
2
votes
1answer
26 views

If $f^p\in L^1([0,1])$ it's bounded a.e.

We know that being Lebesgue integrable does not imply boundedness of the function (e.g. $g(x)=\frac{1}{\sqrt x}$). However function in $L^p$ spaces are functions with some decay conditions. Suppose ...
0
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2answers
23 views

If $f_n + g_n \to h$ in $L^2(\Omega)$ and $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$?

On a bounded domain $\Omega$, if $f_n + g_n \to h$ in $L^2(\Omega)$ and each $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$? I feel like this should be true since each sequence is non-negative, so ...
0
votes
3answers
39 views

Let $f\in L^p$. Can we say $\|f\|_{L^{p}} \leq \epsilon$ on $|x|\geq R$ for large $R$?

Let $f\in L^{p}(\mathbb R), (1\leq p <\infty)$ and $\epsilon>0.$ My Question: Can we expect to find $R>1$ (may be large) so that $\|f\|_{L^{p}(B_R)} \leq \epsilon$ on ...
1
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1answer
32 views

Continuity on $L_p$ spaces

Consider a nonlinear and continuos function $f:\mathbb{R} \rightarrow \mathbb{R}$ and we define the functional \begin{equation} F(u) = \int_{[0,1]^2} f(u(x,y)) dxdy \end{equation} where $u$ is an ...
1
vote
1answer
18 views

Approximations of $L^p$ functions, convolutions, mollifiers, etc. (resource needed)

What is a good resource in which I can read about mollifiers, basic theorems regarding convolutions, smooth approximations of $L^p$ functions and the like? (the presence of exercises would be great, ...
3
votes
1answer
38 views

Proof that $\|S_N\|_p < \infty $ is equivalent to $\|S_N f - f\|_p \to 0$ as $N \to \infty$

I am having difficulties with the proof of proposition 1.9 in the book "Classical and multilinear harmonic analysis, Vol. 1" by C. Muscalu and W. Schlag. The following statements are equivalent ...
1
vote
1answer
26 views

Showing $u'=v$ a.e. given $u_k \to u$ and $u'_k \to v$ in $L_2(\mathbb{R})$.

Suppose $(u_k)$ is a sequence of differentiable functions in $L_2(\mathbb{R})$ satisfying (1) There is a $u \in L_2(\mathbb{R})$ so that $\| u_k - u\|_2 \to 0$. (2) There is a $v \in ...
2
votes
1answer
30 views

Prove that $\rho_n \star f \to f$ in $L^p(R^N)$.

Let $\rho \in L^1(R^N)$ with $\int_{}^{} \rho=1$ .Set $\rho_n(x)=n^N\rho(nx)$. Let $f\in L^p(R^N)$ with $1\leq p<\infty$. Prove that $\rho_n \star f \to f$ in $L^p(R^N)$. My try: Since $f \in ...
1
vote
1answer
32 views

Quotients of $L_1$

I know the rather standard fact in Banach space theory that every separable Banach space is a quotient of $\ell_1$. Is it true that every (possibly non-separable) Banach space is a quotient of some ...
1
vote
1answer
28 views

Subspaces of quotients of $L^p$ spaces

Is the collection of subspaces of quotients of $L^p$ spaces considered to be a large class of Banach spaces?
0
votes
1answer
26 views

Proving that $l_r$ is dense everywhere in $l_p$ $1\leq r \leq p$

$$l_p=\{(x_i)^{\infty}_{i=1}|\sum_{i=1}^{\infty}|x_i|^p<\infty\}$$ The answer is given, but this proof makes no sense to me. If somebody could explain the logic, idea here, I would be very ...
2
votes
0answers
21 views

Volume of n-dimensional ball in L1 norm with change of variables

For a homework problem, I need to find a recursive equation that relates the volume of an $n$-dimensional ball $V_n(r)$ of radius $r$ to that of an $(n-2)$-dimensional ball, expressed by $V_{n-2}(r)$. ...
0
votes
2answers
38 views

Duality of $L^p$ spaces

Let $p,q\in(1,\infty)$ be such that $1/p+1/q=1$ and let $(\Omega, \mathcal A,\mu)$ be a $\sigma$-finite measure space. Claim: The map $$\phi:L^q(\Omega)\to \left(L^p(\Omega) \right)^*,\quad ...
1
vote
1answer
40 views

Boundedness and norm of a sequence operator

Let $s = \{s_{n}\}_{n=1}^{\infty}$ be a fixed and bounded sequence of real numbers, i.e. $s \in (\ell^{\infty},\|\cdot\|_{\infty})$. Consider the operator $T_{s} : \ell^{2} \to \ell^{2}$ defined ...
0
votes
1answer
36 views

$L^2$ and $L^\infty$ normed inequality for PDE solution: Which one is more informative and why?

I have the following inequalities $$max_{t \in [0,T]} \lVert u_1(t, \cdot)-u_2(t, \cdot) \rVert_{L^2(\mathbb{R})} \leq C \lVert g_1(x) - g_2(x) \rVert_{L^2(\mathbb{R})}$$ and $$max_{t \in [0,T]} ...