Intereting Posts

Suppose $f_n(x)=x^n – x^{2n}$ , $x \in $. Dose the sequence of functions $\lbrace f_n \rbrace$ converge uniformly?
Confused by proof in Rudin Functional Analysis, metrization of topological vector space with countable local base
“Direct sums of injective modules over Noetherian ring is injective” and its analogue
inverse element in a field of sets
From distribution to Measure
Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})$
Summation of Fibonacci numbers $F_n$ with $n$ odd vs. even
Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?
Prove that: $2^n < n!$ Using Induction
How to find day of a date
Taylor series for different points… how do they look?
$fg\in L^1$ for every $g\in L^1$ prove $f\in L^{\infty}$
Evaluate the sum of the following Legendre symbols: $\sum\limits_{a\in\mathbb{F}_p \ \text{and} \ a \neq 0,1} \left(\frac{a-a^2}{p} \right)$
What is the equation family of the projectile-motion-with-air-resistance eqn?
Cubic trig equation

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 useful tool, but I dont know how you can transfer it to $\ell^2$? Are there any similar theorems?

- 1-separated sequences of unit vectors in Banach spaces
- Every closed subspace of ${\scr C}^0$ of continuously differentiable funcions must have finite dimension.
- Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)
- Metrizability of weak convergence by the bounded Lipschitz metric
- On Fredholm operator on Hilbert spaces
- Dual Bases: Finite Versus Infinite Dimensional Linear Spaces
- Dense Subspaces: Intersection
- Equivalent Norms on $\mathbb{R}^d$ and a contraction
- Questions about weak derivatives
- Why does the semigroup commute with integration?

There is the following characterization:

$S\subset \ell^2$ is compact if and only the following conditions are satisfies:

- $S$ is closed;
- $S$ is bounded, i.e. $\sup_{x\in S}\lVert x\rVert_{\ell^2}<\infty$;
- we have $\lim_{N\to +\infty}\sup_{x\in S}\sum_{k=N}^{+\infty}|x_k|^2=0$.

- Groups of real numbers
- $f$ such that $\sum a_n$ converges $\implies \sum f(a_n)$ converges
- Very interesting graph!
- Compute $\sum \limits_{k=0}^{n}(-1)^{k}k^{m}\binom{n}{k}$ using Lagrange interpolation.
- The identity cannot be a commutator in a Banach algebra?
- Proving $e^{-|x|}$ is Lipschitz
- Question about a basis for a topology vs the topology generated by a basis?
- No group of order $400$ is simple – clarification
- Proof of $n^{1/n} – 1 \le \sqrt{\frac 2n}$ by induction using binomial formula
- Using permutation matrix to get LU-Factorization with $A=UL$
- Factor $x^4 + 64$
- Representation of integers by Fibonacci numbers
- Differentiable and analytic function
- Is a line parallel with itself?
- Can a periodic function be represented with roots $= x^2$ where $x$ is an element of the integers?