1
vote
0answers
20 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
vote
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?
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 $\{f_{n}\}$ ...
1
vote
0answers
23 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
2
votes
2answers
51 views

is the following set complete in $l_p$?

I'm not sure what is the correct term in English, but by 'complete set' I mean that the span (of finite combinations) of the set is dense in the space. I have to show for which p ($ 1\leq p ...
18
votes
2answers
367 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
33 views

property of a given subset $ A $ of $\mathit{l}^2 $

I was given this exercise Let $\lbrace e_n \rbrace $ be the canonical basis of $\mathit {l}^2 $ (the space of square sommable series with the usual norm) and set $$ A:= \lbrace \sum_{n \ in ...
2
votes
2answers
116 views

Differences between $L^p$ and $\ell^p$ spaces

Could someone explain some differences between the $L^p$ and $\ell^p$ spaces? Thanks a lot.
7
votes
4answers
337 views

$\ell^p$ is not isometric to $\ell^q$

The problem is this: if $1\le p<q<\infty$ then $\ell^p$ and $\ell^q$ are not isometric (as Banach spaces). This is an exercise but I'd like to see an elegant proof.
3
votes
1answer
50 views

Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
0
votes
1answer
127 views

characterization of weakly convergent to zero sequences on $l^p$ for $1\le p < \infty$

Let $1\le p< \infty$. Show that a sequence $t_k = ({t_{kj}})_{j=1}^{\infty}\in l^p$ converges weakly to 0 iff $||t_k||_p$ is bounded and $\lim_k t_{kj}=0$. I proved that if $t_k$ converges weakly ...
1
vote
0answers
129 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
1
vote
1answer
186 views

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like All classical (separable) Banach spaces used in analysis have a Schauder basis . I was wondering where to find a proof confirming ...
1
vote
1answer
291 views

Doubt in the proof that $l^{p}$ is complete

I was looking at the proof that $l^{p}$ is complete with respect to the standard metric. Suppose $x^{(n)}$ is a Cauchy sequence in $l^{p}$. Then Given $\epsilon > 0$, $\exists\,\, n_{0} \in ...
0
votes
2answers
64 views

function in $L^1\setminus L^2$

I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded). Does anyone know such a function?
3
votes
0answers
73 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
5
votes
1answer
144 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
12
votes
1answer
125 views

Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
3
votes
2answers
197 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. $n,m > N$ ...
0
votes
2answers
119 views

How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
1
vote
0answers
440 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
1
vote
1answer
44 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
2
votes
1answer
120 views

Closed subspace of $L^1[0,1]$

The statement I need to prove is following. Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ ...
2
votes
1answer
151 views

weak vs. norm compactness in $\ell_1$

So I'm trying to show that weakly compact sets in $\ell_1$ are norm-compact. I've already proven that weak sequential convergence implies norm convergence. I think the idea I want to go with is to ...
1
vote
1answer
151 views

prove a subset of squence space $l^p$ closed in strong topology

Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space $$S=\{X_{mn}: m,n≥1\}$$ where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and ...
1
vote
1answer
113 views

Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite ...
7
votes
1answer
361 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 ...
3
votes
3answers
505 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 ...
6
votes
1answer
530 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?
12
votes
1answer
673 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.
11
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 ...