# Tagged Questions

49 views

### Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
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### Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
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46 views

### I'm searching for the formula of the series $\sum_{n=0}^{\infty}a^{n^l}$

I'm searching for the sum-formula (if exists) of the following power series: $$\sum_{n=0}^{\infty}a^{n^l}$$ where $l=2,3,....$, and $|a|<1$.
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### Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval. Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ ...
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66 views

### Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
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### family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
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### Sequence spaces

Suppose the sequence spaces $$d \colon=\left\lbrace \left\lbrace x_n\right\rbrace_{n \in \mathbb{N}} \in \mathbb{K}^{\mathbb{N}} \colon x_n=0 \ \text{for almost all} \ n \right\rbrace$$ and ...
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### Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
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### Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
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### $(\ell_p,\|.\|_{\infty})$ 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?
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### Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
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### Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy,$$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
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### Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)}$$ for ...
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### What is the main defferences between nets and ordinary sequences

I know that there are many results in metric spaces (or first-countable topological spaces) can be describe in the language of sequences but these results might not be true in general topological ...
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### Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
### Is the sequence of function $\frac{\sin nx}{\pi x}$ convergent to $\delta(x)$
I have no doubt that the integral of the function from negative infinity to positive infinity is 1. But I don't think $f_n(x) = \frac{\sin nx}{\pi x} \rightarrow 0$ as $n \rightarrow \infty$ given ...
For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that \$\left\| f_a(x)-f_a(y) \right\| \leq ...