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Sobolev Space

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

Cambridge IA Numbers and Sets

A proof is a finite sequence of statements, each of which is either an axiom, a previously established result, or follows from earlier statements in the sequence by a valid logical rule, such that the

Discrete Mathematics 1

Continuity (Real Analysis)

Consider the polynomial p(x) = x^3 - 2. Since p(1) = -1 0, the function changes sign on [1, 2], so there must be a root somewhere in between — namely sqrt[3]2. This reasoning is the Intermediate Value

Analysis 1

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Cantor Set

Analysis

How large can a subset of $$ be while containing no interval? At first glance, removing open intervals from $$ should eventually leave behind a set that is either countable (and hence negligible) or retains some interval of positive length. The Cantor set demolishes both intuitions: it is uncoun

Updated Apr 23

Cambridge II Integrable Systems

Analysis Partial Differential Equations

Integrable systems represent one of the deepest and most beautiful intersections of geometry, dynamics, and analysis in mathematical physics. Rather than viewing these systems as exceptional cases, this course positions integrability as a fundamental organizing principle: many of the nonlinear equat

Updated Apr 22

Cambridge II Galois Theory

Algebra Abstract Algebra

Let $f(t) \in \mathbb{Q}[t]$ and let $X$ be a field extension of $\mathbb{Q}$ (or more generally, any set in which $f$ can be evaluated). The root set of $f$ in $X$ is \begin{align} \mathrm{Root}_f(X) = \{ \alpha \in X : f(\alpha) = 0 \}. \end{align}

Updated Apr 21

Operator Topologies

Analysis

Given a $X$ and a bounded linear operator $T \in \mathcal{L}(X, X)$, the question "does a sequence of operators $T_k$ converge to $T$?" admits no single answer. The operator norm topology — the most natural candidate — demands that $T_k$ approximate $T$ uniformly across the en

Updated Apr 18

Riesz's Lemma

Analysis

In finite-dimensional, the geometry of the unit ball is tightly constrained by compactness. The closed unit ball $\overline{B}(0, 1) = \{x \in X : \|x\| \le 1\}$ in $\mathbb{R}^n$ is compact by the Heine-Borel theorem, and this compactness guarantees that many optimis

Updated Apr 18

Radon-Nikodym Theorem

Analysis

Given a $(X, \mathcal{A}, \mu)$ and a second measure $\nu$ on the same $\sigma$-algebra $\mathcal{A}$, one frequently needs to express integrals with respect to $\nu$ in terms of integrals with respect to $\mu$. For instance, if $\mu = \mathcal{L}^1$ is Lebesgue measure on $\

Updated Apr 18

Archimedean Property

Analysis

Every argument in real analysis that begins "let $\varepsilon > 0$" depends, at its core, on being able to find a natural number large enough to make some quantity small. If we want $1/n < \varepsilon$, we need an $n \in \mathbb{N}$ with $n > 1/\varepsilon$. This is not a logical triviality — it is

Updated Apr 18

Axiom of Choice

Analysis

Throughout mathematics, constructions that feel routine in the finite case become deeply problematic in the infinite case. Given a finite collection of nonempty sets, one can simply pick an element from each --- no principle beyond the logical rules of set theory is needed. But given an infinite f

Updated Apr 18

Weierstrass Approximation Theorem

Analysis

How well can polynomials approximate continuous functions? The question seems straightforward — after all, Taylor's theorem tells us that a smooth function is well-approximated by its Taylor polynomials near any given point. But Taylor approximation is fundamentally local: the polynomial is tuned

Updated Apr 18

Cambridge III Advanced Probability

Probability & Statistics Probability Theory

This course develops the core machinery of modern probability theory, beginning with the measure-theoretic formulation of conditional expectation and culminating in the construction and analysis of Brownian motion and Poisson random measures. The central thread is the interplay between martingale theory and the fine structure of stochastic…

Updated Apr 18

Convergence in Measure

Analysis

In analysis, we frequently encounter sequences of measurable functions $(f_n)$ converging to a limit $f$, and need to pass to the limit in integrals, compositions, or other operations. The most natural notion — pointwise convergence — turns out to be poorly behaved: i

Updated Apr 18

Simple Function

Analysis

The construction of the Lebesgue integral confronts a fundamental design problem: how should one assign a numerical value to the integral of a general measurable function $f: X \to [0, \infty]$ on a measure space $(X, \mathcal{A}, \mu)$? T

Updated Apr 18

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Cantor Set

Cambridge II Integrable Systems

Cambridge II Galois Theory

Operator Topologies

Riesz's Lemma

Radon-Nikodym Theorem

Archimedean Property

Axiom of Choice

Weierstrass Approximation Theorem

Cambridge III Advanced Probability

Convergence in Measure

Simple Function

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Explore Mathematics

Create New Bundle

Browse by Area

Trending This Week

Sobolev Space

Cambridge IA Numbers and Sets

Continuity (Real Analysis)

Recently Updated

All Pages

Cantor Set

Cambridge II Integrable Systems

Cambridge II Galois Theory

Operator Topologies

Riesz's Lemma

Radon-Nikodym Theorem

Archimedean Property

Axiom of Choice

Weierstrass Approximation Theorem

Cambridge III Advanced Probability

Convergence in Measure

Simple Function