# Questions tagged [arzela-ascoli]

The Arzela-Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics. Use this tag alongside (real-analysis).

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### Compactness of integral operators

came across this exercise I haven't been able to solve. I saw a very similar exercise (to prove an integral operator with a kernel is compact), but it is just different enough, I think, to warrant its ...
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### Constructing an RKHS from a Kernel

I'm reading the book "High Dimensional Statistics" by Martin Wainwright just for fun (also as preparation of my PhD in computer science/Machine Learning). In particular, I'm currently ...
1 vote
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### On Schoen and Yau's proof of the positive mass theorem: extracting a minimal surface in the limit

I'm reading Schoen and Yau's 1979 paper on the Positive Mass theorem. I'm having trouble understanding the proof of how they extracted a minimal surface as the limit of solutions to the Plateau ...
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### Solution verification of a proof of the Peano existence theorem, using Arzela-Ascoli

$\newcommand{\o}{\mathcal{O}}\newcommand{\d}{\,\mathrm{d}}$I believe what I was required to show is a general version of the Peano existence theorem. Let $\o\subseteq\Bbb R\times\Bbb R^n$ be open and ...
133 views

### A version of Ascoli-Arzelà using modulus of continuity and nth entropy numbers

The classical Ascoli-Arzelà theorem could be stated as follows: Let $K$ be a compact metric space and let $\mathcal{H}$ be a bounded subset of $C(K)$ - the space of continuous functions over $K$ with ...
436 views

### Sequence of contraction mapping and convergence of fixed point

Let $(𝑆,||_{\infty})$ be a metric space and $𝑇 : 𝑆→𝑆$ be a function mapping S into itself. $S$ is a space of bounded and Lipschitz continuous function. For each $𝑛\inℕ$, $\tau_{n}\in T$ satisfies ...
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1 vote
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### How can we derive the Arzelà-Ascoli theorem for a $\sigma$-compact space from the ordinary Arzelà-Ascoli theorem?

Let $X$ be a compact topological space and $\mathcal F\subseteq C(X,\mathbb C)$. By the ordinary Arzelà-Ascoli theorem, we know that If $\mathcal F$ is relatively compact in the uniform topology, ...
1 vote
### Pointwise convergence of $F_n(t) = \max_{y \in \mathbb R^N} \int_{B_t(y)} |u_n|^2 \ dx$
The following is a claim in some lecture notes I am reading: Consider a bounded sequence $(u_n) \subset H^1(\mathbb R^N)$ with $\|u_n\|_{L^2} \to \lambda > 0$. For $n \in \mathbb N$, the ...
### Set of functions on $C[0,1]$ with bounded derivative is compact
Just want to check if my proof works or not. Question: Given $K > 0$, determine if the set \begin{equation*} Y = \{f \in C^1[0,1], f(0) = 0, \Vert f'\Vert_\infty\leq K\} \end{equation*} is ...