Learn Theoremsv1.0

Compactness Theorem (Gödel-Maltsev)

A set of propositional formulas is satisfiable if and only if every finite subset of it is satisfiable. This bridges finitary logical consequences to infinite …

math.LO · Logic · math.GM

📚 Undergraduate · Logic

Closed Graph Theorem (Banach-Schauder)

A linear operator between Banach spaces is continuous if and only if its graph is a closed subset of the product space. This result, a key legacy of the Open M…

math.FA · Functional Analysis · math.LO

📚 Graduate · Real Analysis

Church's Theorem (Alonzo Church)

First-order logic is undecidable: there is no algorithm to determine the validity of any arbitrary formula. Step through the proof; understand how the arithmet…

math.LO · Logic

📚 Undergraduate · Logic

Cayley's Theorem (Arthur Cayley)

Every group G is isomorphic to a subgroup of the symmetric group Sym(G). By representing group elements as permutations via left multiplication, we embed any a…

math.GR · Group Theory · math.LO

📚 Undergraduate · Algebra

Cauchy's Integral Formula (Cauchy)

Cauchy's Integral Formula expresses a holomorphic function's value as an interior integral. Step through the replay to see how contour deformation re…

math.CV · Complex Variables · math.CA

📚 Undergraduate · Complex Analysis

Cauchy-Goursat Theorem (Cauchy & Goursat)

If f(z) is analytic in a simply connected domain D, its line integral over any simple closed contour in D is zero. Step through the proof; the Goursat approach…

math.CV · Complex Variables · math.CA

📚 Undergraduate · Complex Analysis

Casorati-Weierstrass Theorem (Casorati, Weierstrass)

Near an essential singularity, a holomorphic function takes values dense in the complex plane. This theorem proves that essential singularities are not just po…

math.CV · Complex Variables · math.CA

📚 Graduate · Complex Analysis

Cantor's Theorem (no surjection onto the power set)

For every set A there is no surjection onto its power set, so the power set is strictly larger. Cantor's diagonal argument — the engine behind the uncount…

math.LO · Logic

📚 Undergraduate · Set Theory

Cantor-Schroeder-Bernstein Theorem

If two sets admit injective maps into each other, there exists a bijection between them. A cornerstone of set theory, proving the intuitive notion of size comp…

math.LO · Logic · math.GM

📚 Undergraduate · Logic

Cancellation Law (Group Theory)

In any group, products can be simplified by removing common factors, thanks to the existence of group inverses. The Cancellation Law is the fundamental algebra…

math.GR · Group Theory · math.RA

📚 Undergraduate · Algebra

Brouwer Fixed-Point Theorem (Brouwer)

Every continuous map from the closed unit disk to itself must have at least one fixed point. This result, proven here via the non-existence of a retraction, is…

math.AT · Algebraic Topology · math.GN

📚 Graduate · Topology

Bolzano–Weierstrass Theorem (Bolzano, Weierstrass)

Every bounded sequence of real numbers has a convergent subsequence. The proof uses a bisection method to trap the terms in ever-shrinking intervals. Step thro…

math.CA · Classical Analysis and ODEs · math.FA

📚 Undergraduate · Real Analysis

Baire Category Theorem (René-Louis Baire)

In a complete metric space, the intersection of countably many dense open sets is dense. The bedrock of modern analysis, it links completeness to topological d…

math.GN · General Topology · math.FA

📚 Undergraduate · Topology

The Argument Principle (Cauchy)

The logarithmic derivative of a meromorphic function maps its zeros and poles to simple poles with residues equal to their multiplicities, allowing us to count…

math.CV · Complex Variables · math.CA

📚 Graduate · Complex Analysis

Archimedean Property (Archimedes)

The real numbers have no infinitely large elements; for any positive x and any y, there exists a natural n such that nx > y. Proven via the Completeness Axi…

math.CA · Classical Analysis and ODEs · math.RA

📚 Undergraduate · Real Analysis