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androma.org/page/sobolev-spaces

Sobolev spaces are fundamental function spaces in modern analysis and PDE theory. They extend the classical notion of differentiability by using weak derivatives, which allows functions with limited regularity to be treated within a rigorous functional-analytic framework.

Sobolev Space:

The Sobolev space $W^{k,p}(U)$ consists of all locally summable functions $u: U \to \mathbb{R}$ such that for each multi-index $\alpha$ with $|\alpha| \leq k$, the weak derivative $D^{\alpha}u$ exists and belongs to $L^p(U)$.

$$\|u\|_{W^{k,p}(U)} = \left( \sum_{|\alpha| \leq k} \int_U |D^\alpha u|^p \, dx \right)^{1/p}$$

A key tool in the study of Sobolev spaces is the mollifier, which allows approximation of $L^p$ functions by smooth functions.

Colloq

Maria, Alex, Jake

# analysis

# algebra

# topology

Sophie Park

Tom Nguyen

Rachel Li

# analysis

Maria Chen 2:34 PM

I've been trying to prove that the Sobolev embedding $W^{1,p} \hookrightarrow L^{p^*}$ is compact when restricted to bounded domains. Is the Rellich–Kondrachov theorem the right tool here?

🤔 2

2 replies

Alex Kim 2:36 PM

Yes! Rellich–Kondrachov gives you compactness for the subcritical embedding. The key hypothesis is that your domain has a Lipschitz boundary. Without that regularity you can construct counterexamples.

💡 3 👍 1

Maria Chen 2:38 PM

That makes sense. I'll add the boundary regularity assumption to my proof and push a PR to the Sobolev Spaces page. Thanks!

Jake Liu is typing...

Rachel Li 11:30 AM

Quick question: Is $\mathbb{Z}[\sqrt{-5}]$ a UFD?

Tom Nguyen 11:45 AM

No! Classic counterexample: $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$. Two different factorizations!

🤯 2 👍 3

Sophie Park 9:15 AM

Working through the proof that every compact Hausdorff space is normal. The Urysohn lemma step is beautiful.

❤️ 4

Sophie Park 10:12 AM

Hey, did you see the new Urysohn lemma write-up? The proof sketch on the wiki page is really clean.

You

You 10:14 AM

Yes! I left a suggestion on the PR. The partition of unity argument could be tightened.

Sophie Park 10:15 AM

Good call. Want to hop on a Colloq and talk through it?

You

You Yesterday

Tom, your counterexample in #algebra was great. Do you have a reference for the class number computation?

Tom Nguyen Yesterday

Thanks! Check Marcus, "Number Fields," Chapter 5. The Minkowski bound argument is all you need.

Rachel Li 3:00 PM

Are you coming to the study session tomorrow? We're covering Galois theory Chapter 6.

You

You 3:02 PM

Definitely. I'll prep the splitting field examples. Can we use the #algebra Colloq?

Rachel Li 3:03 PM

Perfect, I'll set it up.

👍 1

Message #analysis...

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REAL ANALYSIS

Theorem

What does Bolzano–Weierstrass guarantee, and why did I need it in the extreme value theorem proof?

main.tex

sections/intro.tex

sections/proofs.tex

references.bib

figures/

Issues 3

Pull Requests 1

Discussion

\usepackage{androma} % cite wiki theorems

\usepackage{hyperref}

\section*{Compactness in Euclidean Space}

We begin by recalling the notion of compactness,

which plays a central role in analysis and topology.

\begin{definition}[Compact Space]

A topological space $X$ is \textbf{compact} if every

open cover of $X$ has a finite subcover.

\end{definition}

In $\mathbb{R}^n$, compactness admits a clean

characterization:

% Pull a theorem directly from the wiki:

\quotetheorem{Heine-Borel}

$Compiled LaTeX preview$

Add Cauchy–Schwarz proof

Open

Maria wants to merge 1 commit into main

Cauchy–Schwarz Inequality.md +6 −0

1414 $|\langle u, v \rangle| \leq \|u\| \cdot \|v\|$

1515

16 [proof]

17 If $v = 0$, the result is trivial. Suppose $v \neq 0$.

18 For any $t \in \mathbb{R}$, we have $\|u + tv\|^2 \geq 0$.

19 Setting $t = -\langle u,v\rangle/\|v\|^2$ yields the result.

20 [/proof]

2 reviewers approved

Fix typo in MVT remarks

Open

Jake wants to merge 1 commit into main

Mean Value Theorem.md +1 −1

2828 [remark:Geometric Interpretation]

29 The MVT says there is always a point on the curve where the tangent

29 The MVT says there is always a point on the curve where the **tangent**

3030 line is parallel to the secant through $(a, f(a))$ and $(b, f(b))$.

1 reviewer approved

Add L'Hopital's Rule

Merged

Alex merged 1 commit into main

L'Hopital's Rule.md +6 −0

1 [theorem:L'Hopital's Rule]

2 If $\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0$ (or $\pm\infty$),

3 and $g'(x) \neq 0$ near $c$, then:

4 $$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$

5 provided the right-hand limit exists.

6 [/theorem]

3 reviewers approved

Why every math student should learn to write proofs collaboratively

By Viktor Miykov · March 15, 2026 · 8 min read

Mathematics has always been a collaborative discipline. From the letters between Euler and Goldbach to the Polymath projects, the best mathematical work emerges from dialogue. Yet most students learn to write proofs in isolation, receiving feedback only from a single grader weeks after submission.

What if proof-writing looked more like software development? A pull request workflow for mathematical content enables peer review, iterative refinement, and attribution tracking — all skills that translate directly to research collaboration. On Androma, we've seen student proof quality improve by 40% when using the review workflow compared to traditional submission.

Education Collaboration Proof Writing

Definition. For an open set $U \subseteq \mathbb{R}^n$, the Sobolev space $W^{k,p}(U)$ consists of all locally summable functions $u : U \to \mathbb{R}$ such that for each multi-index $\alpha$ with $|\alpha| \leq k$, the weak derivative $D^{\alpha}u$ exists and belongs to $L^p(U)$.

TEXT Maria Chen Verified by Alex Kim

The Sobolev norm captures both the size of a function and the size of its derivatives up to order $k$, measured in the $L^p$ sense. This makes $W^{k,p}$ a Banach space, and in the special case $p = 2$, a Hilbert space denoted $H^k$.

TEXT Alex Kim

Theorem (Sobolev Embedding). If $kp < n$, then $W^{k,p}(\Omega)$ embeds continuously into $L^q(\Omega)$ for all $q \leq p^* = np/(n - kp)$. If $kp > n$, then $W^{k,p}(\Omega)$ embeds into $C^{0,\alpha}$ for suitable $\alpha$.

THEOREM Jake Liu Verified by Maria Chen

MC Maria Chen 12 blocks · 48%

AK Alex Kim 8 blocks · 32%

JL Jake Liu 5 blocks · 20%

AK Alex Kim Reviewer

83% verified 20 / 24 blocks

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