5
votes
1answer
37 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
0
votes
1answer
37 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
1
vote
1answer
23 views

Convergence of the sequence of operators on a Banach space.

Let $T$ be a bounded linear operator on a Banach space $E$, and let $(T_n)$ be a sequence of bounded linear operators on $E$ such that $$\| (T - T_n)x \| \to 0 \ \ \text{ as } n \to \infty$$ for all ...
1
vote
1answer
40 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
2
votes
1answer
77 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
1
vote
0answers
29 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 $f:I\mapsto Y$ ...
1
vote
1answer
65 views

Proposed proofs for weak convergence question

I have the following question and two proposed proofs. Please advise if these proofs are adequate and which of the two is better. Thanks. Question: Let $V$ be a reflexive, separable Banach space. ...
0
votes
1answer
17 views

SOT Convergence and Compact Convergence

Let $E$ be a Banach space, and let $A(E)$ denote the closure of the finite rank opertors on $E$. Let $(S_\alpha)$ be a bounded net of operators on $E$ such that $S_\alpha T\to T$ for all $T\in A(E)$. ...
2
votes
0answers
36 views

Are all functions in a Banach space convergent?

Are all functions in a Banach space convergent? I need this answer in a study of wavelet analysis. My thoughts are: since we have this definition: Let $X$ be a Banach space. A sequence of vectors ...
1
vote
0answers
53 views

Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
1
vote
1answer
41 views

Easy question about $C_c^\infty(0,T)$ and $C_c^\infty((0,T);X)$

Let $f \in C_c^\infty(0,T).$ It follows that $f \in C^k(0,T)$ for all $k$, and so if $t_n \to t$ then $$|f(t_n) - f(t)| + |f'(t_n) - f'(t)| + ... +|f^{(k)}(t_n) - f^{(k)}(t)| \to 0$$ for all $k$. Now ...
0
votes
1answer
99 views

Prove that the Set of Bounded Linear Operators is Banach

Let $B(V,V')$ be the vector space formed by set of linear operators $T:V\rightarrow V'$. where $V,V'$ are normed vector spaces. Equip $B(V,V')$ with the norm $$ \|T\|=\sup\frac{\|T(x)\|}{\|x\|} $$ ...
2
votes
1answer
259 views

Closure of the range of a compact operator

Let $X$ be an infinite-dimensional Banach space, and let $Y$ be a banach. Let $T$ be a compact operator from $X$ to $Y$, ie. if $(x_n)$ is a sequence in $X$ then there is a subsequence s.t. ...
2
votes
1answer
159 views

Convergence of functionals and weak convergence

I consider a Banach space $V$ with its dual $V'$. I had a sequence of functionals $\{f_k\}_{k\in \mathbb N} \subset V'$, and I wanted to show (strong or norm) convergence of $f_k \to f \in V'$. I ...
1
vote
1answer
175 views

Show that ($\ell^1$, $\|\cdot\|_1$) is complete

Show that the vector space $\ell^1 : = \{(a_n) : \sum_n|a_n| < \infty\}$ with the norm $\|(a_n)\|_1 : = \sum_n|a_n|$ where $(a_n)$ are sequences in $\mathbb C$ is complete. Thanks in advance.
1
vote
1answer
47 views

The set of finite “variations” of an unconditionally convergent series is pre-compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum_{i=1}^n \varepsilon_ix_i:n\in\mathbb N, \varepsilon_i=\pm1\}$ is pre-compact. Proof: 1) ...
1
vote
1answer
33 views

The set of “variations” of an unconditionally convergent series is compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum \varepsilon_ix_i:\varepsilon_i=\pm1\}$ is compact. Proof: 1) $\{-1,1\}^{\mathbb N}$ is ...
1
vote
1answer
122 views

What is the relationship between convergence uniformly, pointwisely, weakly, in $L^{\infty}$ norm and in $L^{p} $ norm?

What is the relationship between convergence uniformly, pointwise, weakly, in $L^{\infty}$ norm and in $L^{p}$ norm? I am quite puzzled by so many convergences, can anybody tell me what is the ...
1
vote
1answer
189 views

How to show that space is complete?

Let $N_\alpha=\{(x_n)_{n=1}^\infty\mid \sum_{j=1}^n |x_j|\leq Mn^\alpha\}$, where $\alpha\in R$. Show that $N_\alpha$ is Banach space with the norm $\|(x_n)_{n=1}^\infty\|=\sup_{n\in N} n^{-\alpha} ...
0
votes
0answers
30 views

$‎C_{c}(X)$ is a subspace of ‎$‎C_{0}(X)$ ‎but ‎it ‎is ‎not ‎Banach [duplicate]

I want ‎to ‎show ‎that ‎‎$‎C_{c}(X)$ is a subspace of ‎$‎C_{0}(X)$ ‎but ‎it ‎is ‎not ‎Banach.‎‎ I think : I should prove that if ‎$‎‎‎\{f_{n}\}‎‎$ ‎be‎ a‎ ‎sequence ‎in‎ $‎C_{c}(X)‎$ ‎with ...
2
votes
2answers
301 views

$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence

Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence. $C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$. And, the ...
1
vote
2answers
250 views

From weak and weak star to norm convergence

I haven't found this yet and I'm somehow not sure if my idea is correct. The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ ...
1
vote
1answer
97 views

Sequences in Banach spaces [duplicate]

I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be ...
1
vote
2answers
226 views

A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$

For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and ...
1
vote
0answers
59 views

A question about convergence in $L^p$. [duplicate]

Let $E$ be measurable and $1 \le p \le \infty$. Suppose $\{f_n\}_{n \in \mathbb{N}}$ all measurable and $\{f_n\}_{n \in \mathbb{N}} \to f$ pointwise a.e. $E$. For $p$ as above, I want to show that: ...
1
vote
1answer
1k views

About Banach Spaces And Absolute Convergence Of Seires

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
1
vote
0answers
493 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 ...
2
votes
2answers
93 views

A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other

The functionals $$ \phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t $$ define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$. a) Show that $(\phi_n)$ converges ...
3
votes
1answer
273 views

Space Sobolev $W^{m,p}$ complete

Show that Sobolev space is complete. I am trying Than $L^p(\Omega)$ is complete then If $f_n \in L^p(\Omega)$ then $f_n \to f \in L^p(\Omega)$. But rest show that $D^{\alpha}f \in L^p(\Omega)$. How I ...
2
votes
0answers
142 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
2
votes
1answer
200 views

A question regarding convergence of sequences of $L_p$ functions

Let $(X,\mathcal M, \mu)$ be an arbitrary measure space and $1\le p<\infty$. I am curious whether the following statement holds: Let $\{f_n:X\to\mathbb{R}:n\in\mathbb{N}\}_n$ be a sequence in ...
1
vote
1answer
93 views

Characterization of strong minimums with slices.

I am doing a proof of a Lemma that isn't in a book. Let $X$ a Banach space and $\emptyset\not=S\subset X$ closed of $X$. Let $f$ be a lower semicontinuous function bounded below in $S$. I have that ...
8
votes
2answers
405 views

$\ell_1$ and unconditional convergence

Thanks to the Riemann theorem we know that absolute convergence and unconditional convergence are the same for $\mathbb{R}$. In all the Frechet spaces absolute convergence implies unconditional ...
1
vote
2answers
64 views

Is the set $E$ of sequences containing only entries $0$ and $1$ in $(m,\left \| \cdot \right \|_\infty)$ complete?

I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left ...
3
votes
1answer
59 views

Disjointly supported functions

Let $(f_n)_n$ be sequence of real-valued continuous functions on a compact, Hausdorff space $K$ with pairwise disjoint (closed) supports satisfying $$0<\inf_n \|f_n\|\leq \sup_n\|f_n\|<\infty.$$ ...
4
votes
4answers
349 views

Does the completeness of a normed vector space only depend on its topology?

Let $V \space$ be a vector space over $\mathbb{R}$, and $\Vert \cdot \Vert_1$, $\Vert \cdot \Vert_2$ norms over $V$, which generate the same topology. Is it always true that if $v_n$ is a Cauchy ...
1
vote
2answers
173 views

Extended James space

The discussion in Convergence in topologies, especially the comments of GEdgar, led me to another (converse) question concerning convergence. In the paper G. A. Edgar, A long James space, in: ...
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 ...
3
votes
2answers
524 views

Converging series in Banach space

Does someone know if the following is true: Let $\mathbb{X}$ be some arbitrary Banach space. $\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$. Now, if the series ...