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
27 views

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ almost uniformly?

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure How to show or give an counterexample: $f_n\rightarrow f$ almost uniformly. We believe it is false. Since both convergences imply there ...
0
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0answers
20 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
2
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0answers
42 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
0
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1answer
23 views

How do you prove $L^{\infty}$ is a C*-algebra?

If we define on $L^{\infty}$ the essential supremum norm ($\| \|_{\infty}$), then how can I prove this norm is submultiplicative ($\| T_1T_2\|_{\infty}\leq \| T_1\|_{\infty}\|T_2 \|_{\infty}\, \forall ...
6
votes
1answer
292 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" ...
2
votes
1answer
21 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
2
votes
3answers
48 views

If $f\in L^1$ has a compact support and $0 \leq p \leq1$ then $|f|^p\in L^1$

My text proved that If $f\in L^1$ is bounded and $p \geq1$ then $|f|^p\in L^1$ I wanted to prove the seemingly very similar statement: If $f\in L^1$ has a compact support and $0 \leq p ...
1
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1answer
31 views

Is 'f' belong sobolev?

I was trying to show that the function $$f(x) = \dfrac{x^{1/2}}{1+x^2} \in W^{1,3/2} (0,\infty)$$ that is, have to show that $$f\in L^{3/2}(0,\infty)$$ and $$f_x\in L^{3/2}(0,\infty).$$
1
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1answer
59 views

$\ell^p(I)$ space and a dense set of this space

Let $I$ be an infinite set and $1\leq p<\infty$. Show that $\ell^p(I)$ has a dense set of the same cardinality as $I$. For this I put $$X=\{(x_i); x_i\in \Bbb C , x_i=0 \text{ for all but ...
0
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1answer
59 views

Isometry from $\ell^1$ to $\ell^\infty$

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
1
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0answers
21 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
2
votes
0answers
32 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
1
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1answer
25 views

The derivative of a function is square integrable assuming Fourier transform dominated

I am struggling in solving the second part of this problem. Let $g$ be a continuous function in $L^1(\mathbb{R})$ whose Fourier transform is the function $F$. Suppose $|F(x)|\leq (1+x^2)^{-2}$. Prove ...
1
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1answer
16 views

Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
1
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1answer
19 views

How to prove $f(x)=e^{\frac{1}{x}}$ is continous in $(0,a), a>0 $ and $\int_{0}^{a}e^{\frac{y}{x}}dx, y>0$ does not exist

I would aprecciate any advice. I'm trying to prove that in the context of a measure space, $(X,B,\lambda)$ , with $X=(0, + \infty) $, $B$ the Borel sigma-algebra and $\lambda$ the Lebesgue measure, ...
1
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1answer
27 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
0
votes
1answer
44 views

Inequality important in $L^p$ space

If$\,\,$ $0<p<\infty$, put$\,\,$ $\gamma_{p}=\max(1,2^{p-1})$, and show that $$|\alpha-\beta|^p \leq \gamma_{p}(|\alpha|^p + |\beta|^p)$$ for arbitrary complex numbers $\alpha$ and $\beta$. ...
1
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2answers
52 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
-1
votes
1answer
33 views

Show $W^{1,p}(a,b)$ is compactly embedded in $L^p(a,b)$ for any $1<p\leq \infty$

I always get these sorts of questions wrong. Any help would be appreciated. This is my answer: The problem assumes $(a,b)$ is any bounded interval in $\mathbb R$. Consider a bounded sequence ...
1
vote
1answer
18 views

I would like to show that $\ell^1$ is separable

So here is my question, I want to prove that $\ell^1$ is separable. So i need to show that there exists a countable dense subset in $\ell^1$. Since I am not sure if my idea was right i hoped ...
0
votes
1answer
91 views

Show that $L^1$ is strictly contained in $(L^\infty)^*$

How does one show that $L^1$ is strictly contained in $(L^\infty)^*$? Here, $(L^\infty)^*$ is the space of linear continuous functionals on $L^\infty$.
4
votes
3answers
58 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
1
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0answers
16 views

Criteria for a parameter-dependent integral to be squared integrable

I'm currently stumbling over a criteria of integrability. Let's suppose we have a real-valued function $h$ defined on the positive real axis which is integrable. We then define the function $K ...
1
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1answer
20 views

When is this function in Lp?

Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i ...
1
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1answer
122 views

Approximation of $L^\infty$ functions by $\mathcal{D}$ functions?

Let $Q=(0,T)\times\Omega$ with $\Omega$ a bounded domain. I read this: "the inequality holds for all $f \in \mathcal{D}(Q)$, and by approximation, it holds for all $f \in L^\infty(Q)$ since the ...
0
votes
1answer
27 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
1
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0answers
43 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
2
votes
1answer
37 views

Use convergence theorem to prove the limit

Suppose $f:\mathbb{R}\to\mathbb{R}$ is $L^1$ and it is continuous at the origin. Let $g_n(x)=\frac{1}{1+x^2}f(\frac{x}{\sqrt{n}})$. The problem is to prove $$\lim_n\int_0^\infty ...
1
vote
1answer
33 views

Which of the following sets are open (or closed)?

a.) $A:= \{(x_n)_{n\in \mathbb{N}} : x_n \in [0,1] \hspace{2mm}\text{for all}\hspace{2mm} n\in\mathbb{N}\}$ in $(l^\infty, \|\cdot\|_{\infty})$ and b.) $B:= \{f\in C([0,1]) : |f(t)-t|<1 ...
0
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0answers
13 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
0
votes
1answer
35 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
1
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1answer
32 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
-1
votes
1answer
69 views

Hilbert space $L^{2}(0,\pi)$

I wanted to know how I should proceed if I wanted to prove that the closed subspace of $L^{2}(0,\pi)$ generated by {$\sin(kx): k=1,2,...$} coincides with $L^{2}(0,\pi)$. Thanks.
2
votes
1answer
52 views

$\int_E f_n\to\int_E f$ implies $f_n\to f$ pointwise

The problem is to show that for a bounded sequence $\{f_n\}$ in $L^p[a,b]$ that $$ f_n\rightharpoonup f\,\Longleftrightarrow\, \int_E f_n\to\int_E f,\,\, \forall E_\text{(measurable)}\subset [a,b]. ...
3
votes
0answers
35 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
18
votes
2answers
370 views

In $\ell^p$, if an operator commutes with left shift, it is continuous?

Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution. What we have: ...
2
votes
1answer
40 views

Bound for $fg\in L^{1}$ in terms of $f,g \in L^{1} $

Let's assume that $fg\in L^{1}$ with $f\in L^{1}$ and $g\in L^{1}$. Is there a common way to bound $\Arrowvert fg \Arrowvert_{1}$ in terms of the norms $\Arrowvert f \Arrowvert_{1}$ and $\Arrowvert ...
10
votes
2answers
106 views

Finite dimensional subspace of $C([0,1])$

Let $S$ be a subspace of $C([0,1])$, i.e. the continuous real functions on $[0,1]$. Assume that there exists $c>0$ such that $\|f\|_\infty\leq c \|f\|_2$ for all $f\in S$. Then $S$ must be ...
1
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0answers
42 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
3
votes
0answers
27 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
1
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2answers
38 views

$f_{n}\to f$ in $L^{1}\implies \hat{f_{n}}\to \hat{f}$ in $L^{1}$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R .$ Suppose there exists $f_{n}, \in L^{1}(\mathbb R)$ such ...
1
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0answers
19 views

The relationship of L^1(U) and C(U)

Let $U$ be a open set of $\mathbb{R}^n$, C(U) is all continuous functions on U, for example C(0,1), when $U=(0,1)$. And $L^1(U)$ is lp-space where $p=1$. It was said that $L^1(U)$ is the completion ...
2
votes
1answer
33 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
2
votes
2answers
43 views

Subsequence of functions in $L^p$

On a problem sheet we were asked to find a sequence of functions $(f_n)_{n \geqslant 0} \in L^p [0,1]$ such that $\lim_{n \to \infty} ||f_n||_p = 0$ but $\lim_{n \to \infty} f_n (x)$ doesn't exist ...
0
votes
2answers
28 views

1.Convergence in $L^1$

Let f be measurable function such that $||f||_\infty=\infty.$ Show that there exists {${g_ n}$} $\subset L^1$ such that $||fg_n||_1\to\infty$. Anyhelp would be appreciated..
1
vote
1answer
29 views

Approximation in $L^p$ spaces

Let $X$ denotes the span {$x^n:n \ge1 $}. Is it true that $X $ is dense in $L^1([0,1])?.$ I showed that $X$ is dense in the space of continuous functions that vanishes at zero. I also know space of ...
0
votes
1answer
32 views

Norm of operator between two $L^p$ spaces?

In my reading, I've come across notation like $ ||T||_{2 \to \infty} $, where $T$ is an operator defined on every $L^p$ space. What does this mean? Is it simply the norm of $T$ viewed as an operator ...
1
vote
1answer
53 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
0
votes
1answer
27 views

Interpolation inequality in $L^p$ space

Let $f \in L^{p}(E) \cap L^{q}(E)$ with $p<q$. How to prove that $f \in L^{h}(E)$ for every $h \in (p,q)$ and the following interpolation inequality: $||f||_{h} \leq ||f||_{p}^{\frac{p}{h}} + ...
0
votes
0answers
40 views

Inclusion of $L^p$ spaces if $X$ arbitrary

If $X$ is a finite measure space then one can show that if $1 \le p < q$ then $L^q \subseteq L^p$. Is there anything known about the inclusion if $X$ is an arbitrary measure space? Or given some ...