Category:Improvements

• Definitions/Compact Linear Transformations‎ (1 C, 10 P)

• Hölder's Inequality‎ (2 C, 3 P)
• Hölder's Inequality for Integrals‎ (4 P)

• Resolvent Mapping is Analytic‎ (5 P)

The following 197 pages are in this category, out of 197 total.

• Banach-Alaoglu Theorem/Lemma 3
• Banach-Alaoglu Theorem/Lemma 4
• Binomial Theorem Approximations/Examples/Arbitrary Example 1
• Binomial Theorem Approximations/Examples/Arbitrary Example 2
• Bolzano-Weierstrass Theorem/General Form

• Cardinal Equal to Collection of All Dominated Ordinals
• Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum
• Characteristic of Ordered Integral Domain is Zero
• Characteristics of Traversable Graph
• Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
• Definition:Chess
• Chord Length for Regular Polygon
• Circle of Apollonius is Circle
• Circle of Apollonius is Circle/Proof 2
• Definition:Class Interval
• Definition:Classical Propositional Logic
• Compact First-Countable Space is Sequentially Compact
• Definition:Compact Linear Transformation
• Definition:Compact Linear Transformation/Inner Product Space
• Definition:Compact Linear Transformation/Inner Product Space/Definition 1
• Definition:Compact Linear Transformation/Normed Vector Space
• Definition:Compact Linear Transformation/Normed Vector Space/Definition 1
• Compact Subspace of Linearly Ordered Space/Reverse Implication
• Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
• Compactness Theorem/Proof using Henkin Construction
• Composition of Mappings is Associative
• Condition for Ideal to be Total Ring
• Condition for Mapping from Quotient Set to be Injection
• Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample
• Conditions for Strong Minimum of Functional
• Definition:Connection on Manifold
• Definition:Consistent (Logic)/Set of Formulas/Propositional Logic
• Definition:Contravariant Hom Functor
• Definition:Convex Polygon/Definition 3
• Cosine of Sum/Proof 4
• Countable Union of Finite Sets is Countable
• Definition:Covariant Hom Functor
• Cycle does not Contain Subcycles

• Definition:Darboux Function
• De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Forward Implication
• De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice
• Definite Integral from 0 to Half Pi of Even Power of Sine x
• Definite Integral from 0 to Half Pi of Even Power of Sine x/Proof 1
• Definite Integral from 0 to Half Pi of Odd Power of Sine x
• Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 1
• Determinant of Linear Operator is Well Defined
• Dimension of R-Module R is 1
• Dirichlet Integral/Proof 2
• Disjunction has no Inverse

• Eigenvalues of Normal Operator have Orthogonal Eigenspaces
• Empty Set is Well-Ordered
• Epimorphism Preserves Associativity
• Equation for Perpendicular Bisector of Two Points
• Equivalence of Definitions of Compact Linear Transformation
• Equivalence of Definitions of Normal Subgroup
• Equivalence of Definitions of Transitive Closure (Set Theory)
• Equivalence of Subobjects is Equivalence
• Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations
• Euler's Theorem for Planar Graphs
• Definition:Exact Sequence of Modules
• Existential Instantiation/Informal Statement
• Expansion Theorem for Determinants
• Exponential of Sum of Real Numbers/Corollary

• Final Topology with respect to Mapping
• Finite Sequences in Set Form Acyclic Graph
• Fixed-Point Property is Topological
• Fontené Theorems/First
• Definition:Formal Semantics
• Fourth Sylow Theorem/Proof 2
• Fréchet Space (Functional Analysis) is Metric Space

• Gaussian Integers form Euclidean Domain
• Gaussian Integers form Euclidean Domain/Proof 1
• General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution
• General Variation of Integral Functional/Dependent on n Variables
• Generating Function for Elementary Symmetric Function
• Generating Function for Elementary Symmetric Function/Proof 2

• Hermite's Formula for Hurwitz Zeta Function
• Hermitian Matrix has Real Eigenvalues/Proof 2
• Definition:Hilbert Proof System
• Definition:Homogeneous (Model Theory)
• Definition:Hyperbolic Set of Riemannian Manifold
• Hölder's Inequality
• Hölder's Inequality for Integrals

• Image of Small Class under Mapping is Small
• Template:Improve
• Increasing Sequence in Ordered Set Terminates iff Maximal Element
• Increasing Sequence of Sets forms Nest
• Indexed Summation of Multiple of Mapping
• Inequality of Height of Proper Ideal
• Intersection of Ordinals is Smallest
• Definition:Inverse Sine/Real/Arcsine
• Definition:Irreducible (Representation Theory)
• Definition:Irreducible (Representation Theory)/G-Module

• Kernel of Group Action is Normal Subgroup

• L'Hôpital's Rule
• L'Hôpital's Rule/Corollary 2
• Definition:Laurent Series
• Least Upper Bound Property/Proof 1
• Definition:Legendre Transform
• Lexicographic Product of Family of Ordered Sets is Ordered Set
• Limit Ordinal Equals its Union
• Linear Combination of Measures
• Liouville's Theorem (Complex Analysis)
• Liouville's Theorem (Complex Analysis)/Proof 1
• Logarithmic Integral as Non-Convergent Series

• Definition:Magma of Sets
• Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication
• Definition:Measurable Set/Arbitrary Outer Measure
• Medians of Triangle Meet at Centroid/Corollary

• Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite
• Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 1
• Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions
• Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function
• Newton-Girard Identities/Proof 1
• Non-Equivalence as Equivalence with Negation/Formulation 1/Forward Implication/Proof
• Norm on Vector Space is Quasinorm
• Normed Dual Space Separates Points
• Definition:Number Base/Integers
• Number of Arrangements of n Objects of m Types

• Open Sets of Double Pointed Topology/Corollary
• Order Isomorphism iff Strictly Increasing Surjection
• Order Topology equals Dual Order Topology
• Order Topology on Convex Subset is Subspace Topology
• Order-Extension Principle/Strict/Proof 2
• Ordinal Subset is Well-Ordered
• Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral
• Orthonormal Subset of Hilbert Space Extends to Basis
• Outer Jordan Content of Right Triangle

• P-Product Metrics on Real Vector Space are Topologically Equivalent
• P-Product Metrics on Real Vector Space are Topologically Equivalent/Inequality for General Case
• Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order
• Primary Decomposition Theorem
• Prime Magic Square/Examples/Order 3/Smallest
• Primitive of x over Root of a x + b by Root of p x + q
• Product of Divisor Sum and Euler Phi Functions
• Products of 2-Digit Pairs which Reversed reveal Same Product
• Definition:Projection
• Projection from Product Category
• Definition:Proper Relational Structure
• Ptolemy's Theorem
• Ptolemy's Theorem/Proof 2
• Pullback of Quotient Group Isomorphism/Examples/C6 with A4

• Definition:Quotient Module

• Ratio of Consecutive Fibonacci Numbers
• Ratio of Consecutive Fibonacci Numbers/Proof 3
• Rational Numbers with Denominators Coprime to Prime under Addition form Group
• Real Numbers of Type Rational a plus b root 2 form Field
• Reciprocal of Function of Bounded Variation Bounded away from Zero is of Bounded Variation
• Recursively Defined Sequence/Examples/Term is Term of Index less 1 plus 6 times Term of Index less 2
• Relation between Two Ordinals
• Relationship between Transitive Closure Definitions
• Resolvent Mapping is Analytic
• Resolvent Mapping is Analytic/Banach Algebra
• Resolvent Mapping is Analytic/Bounded Linear Operator/Proof 1
• Resolvent Mapping is Analytic/Bounded Linear Operator/Proof 2

• Scheffé's Lemma/Corollary
• Second-Countable Space is First-Countable/Proof 2
• Seifert-van Kampen Theorem
• Definition:Separable Degree
• Definition:Set of Literals
• Definition:Set Partition
• Shape of Cosecant Function
• Shape of Cosine Function
• Shape of Secant Function
• Sigma-Algebra Generated by Finite Partition is Finite Sub-Sigma-Algebra
• Sign of Composition of Permutations
• Sine and Cosine are Periodic on Reals/Corollary/Cosine
• Sine and Cosine are Periodic on Reals/Corollary/Sine
• Smallest Field is Field
• Solution to Legendre's Differential Equation
• Stopped Supermartingale is Supermartingale
• Strictly Increasing Sequence induces Partition
• Strictly Increasing Sequence of Natural Numbers
• Subband of Induced Operation is Set of Subbands
• Subgroup is Subset of Conjugate iff Normal
• Subgroup is Superset of Conjugate iff Normal
• Sufficient Conditions for Uncountability
• Sum of Indexed Suprema
• Sums of Squares of Diagonals of Order 3 Magic Square
• Supremum of Absolute Value of Difference equals Supremum of Difference

• Definition:Taylor Series/Two Variables
• Taylor's Theorem/One Variable
• Taylor's Theorem/One Variable/Proof by Cauchy Mean Value Theorem
• Definition:Tempered Distribution Space
• Total Ordering on Field of Quotients is Unique
• Tribonacci Constant is Unique Real Root of x^3-x^2-x-1
• Definition:True Value

• Union of Chain of Orderings is Ordering

• Volume of Cone is Third of Cylinder on Same Base and of Same Height
• Volume of Cone is Third of Cylinder on Same Base and of Same Height/Proof 2

• World's Hardest Easy Geometry Problem

• Zorn's Lemma

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