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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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Set

Discrete Mathematics Set Theory

Every mathematical structure rests on the same primitive foundation: the set. Groups are sets with a binary operation; topological spaces are sets equipped with a collection of designated subsets; measure spaces are sets with a $\sigma$-algebra and a measure. When a mathematician writes "$f: X \to Y$ is continuous," the objects $X$, $Y$, and $f$…

Updated Mar 9

Center Manifold Theorem

Calculus Differential Equations

In the analysis of dynamical systems, the stability of an equilibrium point $x^$ of $\dot{x} = f(x)$ is typically determined by linearisation: compute the Jacobian $Df(x^)$, find its eigenvalues, and apply the Hartman–Grobman theorem. If all eigenvalues have nonzero real part — the hyperbolic case — the nonlinear flow near $x^$ is topologically…

Updated Mar 9

Second-Order Parabolic Equations

Analysis Partial Differential Equations

The theory of second-order elliptic equations concerns equilibrium states — solutions to $Lu = f$ that do not change in time. In many physical systems, however, the quantity of interest (temperature, concentration, probability density) evolves toward equilibrium, and the transient behaviour is itself the object of study. This leads to parabolic…

Updated Mar 9

Normed Vector Space

Analysis Algebra Functional Analysis Linear Algebra

Analysis requires a way to measure the "size" of the objects it studies — functions, sequences, operators, distributions. A norm is the abstraction of size: it assigns to each element of a vector space a non-negative real number satisfying the triangle inequality, positive-definiteness, and homogeneity. Equipped with a norm, a vector space becomes…

Updated Mar 9

Topological Dual

Analysis Functional Analysis

Every vector space $X$ has a set of linear functionals $f: X \to \mathbb{C}$ — the algebraic dual $X^\#$. But the algebraic dual is too large to be useful: it contains pathological functionals constructed by the axiom of choice that ignore the topology of $X$ entirely. In analysis, one works instead with the topological dual $X'$ (or $X^$): the…

Updated Mar 9

Fourier Stieltjes Transform

Analysis Harmonic Analysis

The Fourier transform assigns to each integrable function $f \in L^1(\mathbb{R}^n, \mathcal{L}^n)$ a continuous function $\hat{f}$ on frequency space. But many of the most important objects in analysis and probability — point masses, singular distributions of random variables, spectral measures of stochastic processes — are not functions at all.…

Updated Mar 7

Topological Vector Space

Analysis Geometry Functional Analysis Topology

In a normed space $(X, \|\cdot\|)$, the topology is determined by a single quantity — the norm — and convergence means $\|x_n - x\| \to 0$. This is the setting for Banach space theory: the linear operators page and the adjoint page develop their theory entirely within this framework. But many of the most important spaces in analysis are not…

Updated Mar 6

Self-Adjoint Operators

Analysis Functional Analysis

The adjoint page developed the construction $T \mapsto T^$ for bounded operators between Hilbert spaces and established the defining identity $(Tx, y)_K = (x, T^y)_H$. A natural question emerges: what happens when an operator equals its own adjoint? The condition $T = T^$ — self-adjointness — is the infinite-dimensional analogue of the symmetry…

Updated Mar 6

The Adjoint of an Operator

Analysis Functional Analysis

Given a linear map $A: \mathbb{R}^n \to \mathbb{R}^m$ represented by a matrix, the transpose $A^\top$ satisfies $(Ax) \cdot y = x \cdot (A^\top y)$ for all $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$. This simple identity — the ability to "move" the operator from one side of the inner product to the other — is the foundation of duality in linear…

Updated Mar 6

Conformal Maps

Analysis Complex Analysis

A central theme in complex analysis is that holomorphic functions with nonvanishing derivative preserve angles between curves. This geometric property — conformality — transforms the study of holomorphic mappings from a purely analytic enterprise into a geometric one: questions about function theory become questions about the shapes of domains,…

Updated Mar 6

Cauchy Transform

Analysis Complex Analysis

The Cauchy integral formula recovers a holomorphic function from its boundary values: if $f$ is holomorphic inside a simple closed contour $\gamma$ and continuous up to the boundary, then \begin{align} f(z) = \frac{1}{2\pi i} \oint_\gamma \frac{f(w)}{w - z} \, dw \qquad \text{for every } z \text{ inside } \gamma. \end{align} The right-hand side…

Updated Mar 6

Branch Cuts

Analysis Complex Analysis

In real analysis, the equation $e^y = x$ has a unique solution $y = \ln x$ for each $x > 0$: the real exponential is a bijection from $\mathbb{R}$ onto $(0, \infty)$, and its inverse is the natural logarithm. In complex analysis, the situation is fundamentally different. The complex exponential $\exp : \mathbb{C} \to \mathbb{C} \setminus \{0\}$ is…

Updated Mar 4

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Center Manifold Theorem

Second-Order Parabolic Equations

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Topological Vector Space

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The Adjoint of an Operator

Conformal Maps

Cauchy Transform

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Explore Mathematics

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Trending This Week

Sobolev Space

Cambridge IA Numbers and Sets

Continuity (Real Analysis)

Recently Updated

All Pages

Set

Center Manifold Theorem

Second-Order Parabolic Equations

Normed Vector Space

Topological Dual

Fourier Stieltjes Transform

Topological Vector Space

Self-Adjoint Operators

The Adjoint of an Operator

Conformal Maps

Cauchy Transform

Branch Cuts