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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16
votes
2answers
5k views

Limit of $L^p$ norm

Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
24
votes
2answers
7k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
10
votes
2answers
1k views

“Scaled $L^p$ norm” and geometric mean

The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ ...
12
votes
1answer
1k views

Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
3
votes
1answer
2k views

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

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 $ ...
4
votes
3answers
562 views

When $L_p = L_q$?

As we know that $L_p \subseteq L_q$ when $0 < p < q$ for probability measure, I was wondering when $L_p = L_q$ is true and why. Is it to impose some restriction on the domain space? Thanks!
12
votes
1answer
725 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
10
votes
4answers
842 views

Convergence of integrals in $L^p$

Stuck with this problem from Zgymund's book. Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < ...
6
votes
1answer
588 views

How do you prove that $\ell_p$ is not isomorphic to $\ell_q$?

I guess that for all $1\le p,q<\infty $, such that $p\ne q$ , the spaces $\ell_p$ and $\ell_q$ are not isomorphic, but how do you prove this?
2
votes
1answer
295 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
1answer
850 views

$\ell_p$ is Hilbert space if and only if $p=2$

Can anybody please help me to prove this.. Let p greater than or equal to 1,show that the space of all p-summable sequences is an inner product space if and only if p=2
4
votes
2answers
288 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. $n,m > N$ ...
1
vote
1answer
176 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 ...
1
vote
1answer
141 views

Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
7
votes
1answer
619 views

Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

In Lieb and Loss's Analysis, I saw that they mentioned $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the ...
3
votes
1answer
174 views

When is a subset of $\ell^2$ compact?

I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a ...
10
votes
1answer
449 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 $y \in K$ ...
8
votes
1answer
536 views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
4
votes
1answer
671 views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
3
votes
1answer
163 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
2
votes
1answer
108 views

How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..

can anyone help me show that if $f\in L^p(a, b)$ then $$ \lim_{t\to 0^{+}}\int_{a}^b|f(x+t)-f(x)|^p\ dx=0.$$ Thanks, any help will be useful..
7
votes
2answers
182 views

What does the $L^p$ norm tend to as $p\to 0$?

This is something I was thinking about, so I'm going to post it as a question and post my own answer. I hope that anyone who wants will comment, correct me if I'm wrong, and add their own knowledge ...
4
votes
1answer
237 views

Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, ...
4
votes
3answers
560 views

Cesàro operator is bounded for $1<p<\infty$

The Cesàro operator $T\colon \ell_{p}\to\ell_{p}$ is defined by $(Tx)_{k}=\frac{1}{k}\sum_{j=1}^{k}x_{j},\: k\in\mathbb{N}$, where $x=(x_{k})_{k=1}^{\infty}$ Show that $T$ is bounded if ...
3
votes
1answer
234 views

Pointwise a.e. convergence implies strong convergence?

Let $ 1 \leq p_1 < p_2 < \infty$, and suppose that $f_n$ is a sequence of functions in $L^{p_1}[a,b]$ such that $f_n \to f$ pointwise a.e. on $[a,b]$. Suppose in addition that $ ||f_n||_{p_2} ...
2
votes
2answers
79 views

What is $L^p$-convergence useful for?

Why do people care about $L^p$-convergence $f_n \rightarrow f$? Are there any interesting application of $L^p$-convergence? For example, if $p=\infty$, then the limit $f$ of the sequence $f_n$ of ...
2
votes
1answer
188 views

$\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
2
votes
2answers
356 views

Proving that the smooth, compactly supported functions are dense in $L^2$.

I have two problems, one of which depends on the other. (1) I want to prove, cleanly (without too much heavy-weight machinery) that, for some (see (2)) set $\Omega \subseteq \mathbb{R}^n$, the space ...
2
votes
1answer
88 views

How can I give a bound on the $L^2$ norm of this function?

I came across this question in an old qualifying exam, but I am stumped on how to approach it: For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let $$(Vf)(x) = ...
5
votes
2answers
2k views

Smooth functions with compact support are dense in $L^1$

Here is another homework question that I did and I'd be glad if you could tell me if it's right. We now strengthen the result of Question Two for $R$ where we have the notion of differentiability. ...
3
votes
1answer
66 views

Is $C^\infty_0$ dense in $C^\infty$ w.r.t. $\|\cdot\|_{L^p}$ and $\|\cdot\|_{W^{1,p}}$?

Is the space $C^\infty_0(\Omega)$ of smooth functions with compact support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
3
votes
1answer
115 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
141 views

Limit of consecutive Lp norms [duplicate]

I've been wrestling with the following proof off and on for a number of days, and I'm in need of a nudge in the right direction. Let $(E,\mathcal{M},\mu)$ be a measure space with $0 < \mu(E) < ...
1
vote
1answer
182 views

Prove that $f(x)$ is integrable on $\mathbb{R}$.

Suppose $f(x)$,$xf(x)$ $\in$ $L_2(\mathbb{R})$. Prove that $f(x)\in$ $L_1(\mathbb{R})$.
1
vote
1answer
86 views

$\lim_{n\rightarrow \infty}\int_0^1f_nhdm=\int_0^1fhdm$, prove $f\in L^p(m)$ , where $1\le p<\infty$.

On $[0,1]$, suppose $\|f_n\|_p\le 1$, $\lim_{n\rightarrow \infty}\int_0^1f_nh\, dm=\int_0^1fh\, dm$, for any $h\in L^\infty(\mu)$, I need to prove $f\in L^p(\mu)$ , where $1\le p<\infty$. I ...
0
votes
1answer
127 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 $L^\infty$.
5
votes
1answer
71 views

An inequality of $L^p$ norms of linear combinations of characteristic functions of balls

Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists ...
2
votes
2answers
201 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.
7
votes
1answer
383 views

Isometry between $L_\infty$ and $\ell_\infty$

It is known that there exist some isomorphism between $L_\infty$ and $\ell_\infty$, which is not explicit at all. Could someone tell me whether there exist an isometric isomorphism between ...
5
votes
2answers
238 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
4
votes
1answer
253 views

$L^2$ norm inequality

I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ ...
3
votes
1answer
156 views

If $f \in L^{\infty}$ and $\exists r < \infty$ so that $\|f\|_r < \infty$, show $\lim_{p \rightarrow \infty} \|f\|_p = \|f\|_{\infty}$ [duplicate]

Question: This is the last part of a 5 part question I am working on. Let $(X,\mu)$ be a possibly infinite measure space. Assume $\exists r < \infty$ with $\|f\|_r < \infty$ and that $\|f ...
3
votes
2answers
713 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
2
votes
2answers
366 views

Proof of Sobolev Inequality Theroem

I have a short question about the proof of Theorem 2 below. I have included Theorem 1's statement since it is used in the proof of Theorem 2. Definition: If $1 \leq p < n$, the Sobolev Conjugate ...
2
votes
1answer
744 views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
9
votes
2answers
146 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 ...
6
votes
1answer
255 views

pythagoras theorem for $L_p$ spaces

Let's consider $L_2(\mathbb{R}^n)$. Let $Y$ be a non empty closed subspace of $L_2(\mathbb{R}^n)$. Let $x\notin Y$. Let $y^*$ be the best approximation of $x$ on $Y$, i.e., $\|x-y^*\|_2=\inf_{y\in ...
5
votes
1answer
139 views

If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$, i.e., $$\int f_i(x_i)\, ...
4
votes
2answers
429 views

Monotonicity of $\ell_p$ norm

Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have $$ \|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p. $$ I have two questions about the above inequality. $(\bf ...
3
votes
1answer
38 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).