# Tagged Questions

34 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 ...
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 ...
36 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 ...
77 views

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 ...
289 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 ...
244 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 $\lambda_k \to \lambda$ ...
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 ...
224 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 ...
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: ...
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, ...
484 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 ...
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 ...
260 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 ...
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 ...
196 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 ...
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 ...
### $\ell_1$ 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 ...