Difference between revisions of "Category:Theorems"

A

• ► Abstract Algebra Theorems‎ (2 C, 21 P)

C

• ► Category Theory Theorems‎ (1 P)

E

• ► Elementary Probability Theorems‎ (9 P)
• ► Elementary Set Theory Theorems‎ (18 P)

F

• ► Functional Analysis Theorems‎ (42 P)

G

• ► Group Theory Theorems‎ (7 P)

L

• ► Linear Algebra Theorems‎ (19 P)

M

• ► Measure Theory Theorems‎ (39 P)
• ► Metric Space Theorems‎ (36 P)
• ► Module Theory Theorems‎ (3 P)

P

• ► Probability Theorems‎ (10 P)

R

• ► Real Analysis Theorems‎ (26 P)

R cont.

• ► Ring Theory Theorems‎ (1 P)

S

• ► Set Theory Theorems‎ (1 C, 25 P)
• ► Statistics Theorems‎ (10 P)

T

• ► Topology Theorems‎ (1 C, 102 P)

1

• UTLOC:1

A

• A cap (B-C) = (A cap B) - C
• A collection of subsets is a sigma-algebra if and only if it is both a p-system and a d-system
• A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections
• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell
• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell/Statement
• A continuous map induces a homomorphism between fundamental groups
• A continuous map induces a homomorphism on fundamental groups
• A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (with equiv. conditions)
• A function is continuous if and only if the pre-image of every basis element is open
• A linear map is injective if and only if its kernel is trivial
• A linear map is injective if and only if the image of every non-zero vector is a non-zero vector
• A linear map is injective if and only if the kernel contains only the zero vector
• A map from two sigma-algebras, A and B, is measurable if and only if for some generator of B (call it G) we have the inverse image of S is in A for every S in G
• A map is continuous if and only if each point in the domain has an open neighbourhood for which the restriction of the map is continuous on
• A map is continuous if and only if the pre-image of every closed set is closed
• A monotonically increasing sequence bounded above converges
• A pair of identical elements is a singleton
• A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
• A proper vector subspace of a topological vector space has no interior
• A sequence consisting of the nth terms of the sequences in a Cauchy sequence of elements in any little-L space is itself a Cauchy sequence of complex numbers
• A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
• A set is dense if and only if every non-empty open subset contains a point of it
• A set is open if and only if every point in the set has an open neighbourhood contained within the set
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself/Statement
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
• A subspace of a Hausdorff space is a Hausdorff space
• A subspace of a Hausdorff space is Hausdorff
• A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
• A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
• A-(A-B) = A cap B
• Addition of Poisson distributions
• An open ball contains another open ball centred at each of its points
• An open set is a neighbourhood to all of its points
• Axiom of completeness
• Axiom of completeness/Statement

B

• Basis criterion (topology)
• Basis for a topology
• Basis for the tensor product
• Borel sigma-algebra
• Borel sigma-algebra generated by
• Borel sigma-algebra of the real line

C

• C( )0,1( ,R) is not complete when considered with the L^1 norm
• Canonical injection of the subspace topology
• Cauchy criterion for convergence
• Cauchy-Schwarz inequality
• Cauchy-Schwarz inequality for inner product spaces
• Task:Characteristic property of the coproduct topology
• Characteristic property of the direct product module
• Characteristic property of the direct sum module
• Characteristic property of the disjoint union topology
• Characteristic property of the disjoint union topology/Statement
• Characteristic property of the product topology
• Characteristic property of the product topology/Statement
• Characteristic property of the quotient topology
• Task:Characteristic property of the subspace topology
• Characteristic property of the subspace topology
• Characteristic property of the subspace topology/Statement
• Characteristic property of the tensor product
• Characteristic property of the tensor product/Statement
• Class of sets closed under complements properties
• Commutativity of intersection
• Compact-to-Hausdorff theorem
• Compactness/Uniting covers proof
• Comparison test for real series

C cont.

• Comparison test for real series/Statement
• Composition of continuous maps is continuous
• Composition of measurable maps is measurable
• Conditions for a Dynkin system to be a sigma-algebra
• Conditions for a generated Dynkin system to be a sigma-algebra
• Conditions for a map to be a measurable map
• Continuity definitions are equivalent
• Continuous map/Claim: continuous iff continuous at every point

D

• De Morgan's laws
• Deriving the exponential distribution from the Poisson distribution
• Deriving the exponential distribution from the time between event in a Poisson distribution
• Discrete metric and topology/Summary
• Distributivity of intersections across unions
• Dynkin system generated by
• Dynkin system/Proof that definitions 1 and 2 are equivalent

E

• Epsilon form of inequalities
• Equivalence classes are either equal or disjoint
• Equivalence of Cauchy sequences/Proof
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/1 implies 2
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/2 implies 3
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/3 implies 4
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/4 implies 1
• Equivalent conditions to a map being a quotient map
• Equivalent conditions to a set being bounded
• Equivalent conditions to a set being bounded/Statement
• Equivalent conditions to a set being saturated with respect to a function
• Equivalent conditions to a set being saturated with respect to a map
• Equivalent formulas
• Task:Equivalent properties to homeomorphism
• Equivalent statements to a set being dense
• Equivalent statements to compactness of a metric space
• Equivalent statements to compactness of a metric space/Statement
• Euclidean norm
• Every continuous map from a non-empty connected space to a discrete space is constant
• Every convergent sequence is Cauchy
• Every lingering sequence has a convergent subsequence
• Every map from a space with the discrete topology is continuous
• Every sequence in a compact space is a lingering sequence
• Every subspace of a Hausdorff space is Hausdorff
• Every surjective map gives rise to an equivalence relation
• Exists functor
• Expectation of the geometric distribution
• Extending pre-measures to measures
• Extending pre-measures to outer-measures

F

• Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
• First group isomorphism theorem
• For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal
• For any vector subspace of a Hilbert space the orthogonal complement and the closure of that subspace form a direct sum of the entire space
• Function factorisation
• Function factorisation theorem

G

• Geometric series
• Given a Hilbert space and a non-empty, closed and convex subset then for each point in the space there is a closest point in the subset
• Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself
• Given a topological manifold of dimension 2 or more and points p1, p2 and q where q is neither p1 nor p2 then a path from p1 to p2 is path-homotopic to a path that doesn't go through q
• Given two open balls sharing the same centre but with differing radius then the one defined to have a strictly smaller radius is contained in the other
• Greater than or equal to/Epsilon form
• Group
• Group factorisation theorem
• Group homomorphism theorem

H

• Homeomorphic topological spaces have isomorphic fundamental groups/Statement
• Homotopy invariance of loop concatenation
• Homotopy invariance of path concatenation
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
• Homotopy of maps is an equivalence relation

I

• If A is a logical consequence of Gamma then the formula set of Gamma union the negation of A is not satisfiable

I cont.

• If a real series converges then its terms tend to zero
• If a set is a neighbourhood to all of its points then it is open
• If a subsequence of a Cauchy sequence converges then the Cauchy sequence itself also converges
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
• If an inner product is non-zero then both arguments are non-zero
• If the composition of two functions is a bijection then the initial map is injective and the latter map is surjective
• If the intersection of two open balls is non-empty then for every point in the intersection there is an open ball containing it in the intersection
• If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
• Image of a connected set is connected
• Index of elementary set theory equalities
• Induced homomorphism on fundamental groups
• Integral of a positive function (measure theory)
• Intermediate value theorem
• Intersection is commutative

L

• Law of total probability
• Lebesgue number lemma

M

• Mdm of a discrete distribution lemma
• Mdm of the Poisson distribution
• Mean value theorem
• Measures are monotonic and subtractive
• Metric topology
• Module factorisation theorem
• Monotone convergence theorem for non-negative numerical measurable functions
• Monotone convergence theorem for non-negative numerical measurable functions/Statement
• Monotonicity of the integral of non-negative extended-real-valued measurable functions with respect to a measure

N

• Negation of implies
• Negation of logical and

O

• Operations on convergent sequences of real numbers
• Ordered pair
• Overview of the group isomorphism theorems

P

• Passing to the infimum
• Passing to the infimum/Statement
• Passing to the quotient (function)
• Passing to the quotient (topology)
• Passing to the quotient (topology)/Statement
• Pasting lemma
• Pre-image sigma-algebra
• Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra
• Pre-measure/Properties in common with measure
• Product topology
• Products and coproducts of groups
• Proof that the fundamental group is actually a group
• Proof that the fundamental group is actually a group/Outline
• Proof that the symmetric group is actually a group
• Properties of classes of sets closed under set-subtraction
• Properties of the pre-image of a map
• Pullback norm

R

• Reverse triangle inequality
• Rewriting for-all and exists within set theory
• Ring
• Ring generated by
• R^n is a topological vector space

S

• Semi-ring of half-closed-half-open intervals
• Set subtraction
• Sigma-algebra
• Sigma-algebra generated by
• Square lemma (of homotopic paths)

T

• The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set
• The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement
• The basis criterion (topology)
• The basis criterion (topology)/Statement
• The canonical injections of the disjoint union topology are topological embeddings
• The closure of a linear subspace of a normed space is a linear subspace
• The composition of continuous maps is continuous
• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
• The domain, range and field of a relation exist