Fubini's Theorem Fubini定理
For an absolutely integrable function on a product of sigma-finite measure spaces, the product integral equals either iterated integral.
Mathematics Encyclopedia
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Functor 函子
A functor assigns objects and morphisms between categories while preserving identities and composition.
Fundamental Group 基本群
The fundamental group records loops that cannot be contracted while their basepoint stays fixed.
Fundamental Theorem of Arithmetic 算术基本定理
Every positive integer greater than one has a unique prime factorization, apart from the order of the factors.
Fundamental Theorem of Calculus 微积分基本定理
For continuous integrands, differentiation recovers accumulation and antiderivatives evaluate definite integrals.
Galois Extension Galois 扩张
A Galois extension is an algebraic field extension that is both normal and separable.
Gödel's First Incompleteness Theorem
A consistent effectively axiomatized theory extending Robinson arithmetic cannot decide every sentence. Compare Gödel's omega-consistency argument with Rosser's refinement to separate the hypotheses used in the two proofs.
Graphon
A symmetric measurable kernel represents a dense graph limit. Compare step graphons with sampled random graphs, then study how cut distance captures convergence without fixing vertex labels.
Greatest Common Divisor of Integers 整数的最大公因数
The greatest common divisor generates the ideal of all integer linear combinations of two integers.
Grothendieck Duality
Grothendieck duality relates derived pushforward to a twisted inverse-image functor and dualizing complexes. Use the stated theory for finite type separated schemes over a Noetherian base to organize the proper-map duality formula and its smooth proper specialization.
Grothendieck Site
A site specifies which families of maps count as coverings, allowing sheaf methods to work beyond open subsets of a space. Begin with the covering axioms and the Zariski example before moving to sites of schemes.
Grothendieck Topos
A Grothendieck topos collects the sheaves on a site into a geometric setting in its own right. Read the definition together with morphisms of topoi and the punctual example to see how the sheaf category carries more flexible maps than a chosen site presentation.
Group 群
A group is a set with an associative operation, an identity, and inverses for every element.
Group Action 群作用
A group action assigns compatible permutations of a set to group elements.
Hahn–Banach Extension Theorem Hahn-Banach定理
A bounded scalar linear functional on a subspace of a normed space extends to the whole space without increasing its norm.
Hall’s Marriage Theorem Hall定理
A finite bipartite graph has a matching saturating one side exactly when every subset of that side has enough distinct neighbors.
Halting Problem
There is no algorithm deciding whether every encoded machine halts on its supplied input. Follow the diagonal construction carefully: a machine's own description becomes the input that defeats the proposed decision procedure.
Hamiltonian Floer Homology Hamilton 系统周期解的 Floer 同调
Hamiltonian Floer homology converts periodic Hamiltonian trajectories into generators of a chain complex.
Hensel's lemma 亨泽尔引理
A simple root over the residue field lifts to a root in a complete discrete valuation ring. Follow Newton's successive corrections, then compare the coprime-factor lifting form of Hensel's lemma.
Higher Categories and the Topos Bridge 高阶范畴
Higher categories retain coherent higher morphisms, providing the setting for space-valued presheaves and infinity-topoi.
Hilbert Class Field 希尔伯特类域
The ordinary Hilbert class field is the maximal abelian extension unramified at finite places and split at real places, with Galois group canonically identified with the ideal class group.
Hilbert Space 希尔伯特空间
A Hilbert space is an inner-product space complete in its induced norm.
Holomorphic Function 全纯函数
A holomorphic function is complex differentiable throughout an open subset of the complex plane.
Homotopy of Maps 同伦
Homotopy is a continuous deformation from one map to another, with the domain and codomain fixed.
Ideal Class Group 数域的理想类群
The ideal class group identifies fractional ideals that differ by multiplication by a field element and measures the failure of ideals to be principal.