Database

How does this work?

• Click specific definition and you will see the definition(maybe many equivalent definition, or different setups under different situations), definitions have links to every theorems covered by this database.
• Click specific book name, you will see every theorems, propositions, corollaries, and lemmas, along with their proves, which have keywords specified, leading to links of definition, as well as the references to some theorems proved by this theorem or sort of application of the one you currently see.

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Definitions

• Basic Properties
  • Continuous
  • Bounded
  • Convergence
  • Separate
  • Additive
  • Measure
  • Measurable
  • Cone
  • Equivalent Relations
  • Closed
  • Open
  • Interior
  • Closure
  • Compactness
  • Dense
  • Operators
  • Conjugate
  • Dual
  • Connect
  • Locally properties
  • Monotoneness
  • Group
  • Ring
  • Field(Algebra)

From Books

• Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis
  • Chapter 1
    • Dual Space and Norm in Dual
    • Hyperplane
    • Norm in Bidual
    • Canonical Injection
    • Annihilator
    • Domain
    • Epigraph
    • Convex(of functions)
    • Conjugate(of functions)
    • Young’s Inequality
  • Chapter 2

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• Convex Analysis, Rockafellar RT
  • Chapter 1
    • Section 1
      • Affine Sets
      • Translate(of affine sets)
      • Parallel(of affine sets)
      • Dimension(of affine sets)
      • Hyperplane
      • Orthogonal Complement(of affine sets)
      • Affine Hull
      • Affinely Independent
      • Affine Transformation
      • Orthogonal Complement(of affine transformation)
      • Tucker Representation
    • Section 2
      • Convex(of sets)
      • Half Spaces
      • Convex Combinations
      • Convex Hull
      • Polytope
      • Simplex
      • Vertex of Simplex
      • Midpoint(barycenter) of Simplex
      • Cone
      • Convex Cone
      • Orthant(Nonnegative and Positive)
      • Ray
      • Normal(w.r.t. convex sets)
      • Barrier Cone
    • Section 3
      • Symmetric(of sets)
      • Lattice
      • Complete Lattice
      • Inverse Addition(of sets)
      • Umbra and Penumbra
    • Section 4
      • Concave Function
      • Affine Function
      • Effective Domain
      • Proper Convex Function
      • Improper Convex Function
      • Indicator Function
      • Support Function
      • Gauge Function/ Minkowski Functional
      • Euclidean Distance Function
    • Section 5
      • Infimal Convolution
  • Chapter 2
    • Section 6
      • Euclidean Unit Ball
      • Closure(of sets)
      • Interior
      • Relative Interior
      • Relative Boundary
      • Relatively Open
    • Section 7
      • Lower Semi Continuous Hull
      • Closure(of convex functions)
      • Closed(of convex functions)
    • Section 8
      • Direction(of closed half-lines)
      • Recession Cone/Asymptotic Cone
      • Linearity Space
      • Linearity
      • Rank
      • Recession Function
      • Level Sets
    • Section 9(None)
    • Section 10
      • Locally Simplicial
      • Lipschitz Condition
      • Equi-Lipschitzian
  • Chapter 3
    • Section 11
      • Relative Boundary Point(of convex sets)
    • Section 12
      • Vertical Affine Function
      • Upper Affine Function
      • Lower Affine Function
      • Pointwise Supremum
      • Conjugate(of convex functions)
      • Frenchel’s Inequality
      • Partial Affine Function
      • Partial Quadratic Convex Function
      • Elementary Partial Quadratic Convex Function
      • Symmetric(of functions)
      • Monotone Conjugate(of convex functions)
    • Section 13
      • Co-Finite(of convex functions)
    • Section 14
      • Polar(of convex cones)
    • Section 15
      • Polar(of gauge functions)
      • Minkowski Metric(on R^n)
      • Gaugelike(of functions)
      • Observe(of functions)
    • Section 16(None)
  • Chapter 4
    • Section 17
      • M-Dimensional Skew Orthant
    • Section 18
      • Face(of convex sets)
      • Extreme Points(of convex sets)
      • Extreme Rays(of convex cones)
      • Extreme Directions
      • Exposed Face(of convex sets)
      • Exposed Point(of convex sets)
      • Exposed Direction(of convex sets)
      • Exposed Ray(of convex sets)
    • Section 19
      • Polyhedral convex set(in R^n)
      • Finitely Generated Convex Set(FGCS)
      • Generator(of FGCS to generate cones)
      • Generalized Polytope
      • Polyhedral Convex Function
      • Finitely Generated Convex Function(FGCF)
    • Section 20(None)
    • Section 21
      • Consistent(of a system)
      • Incosistent(of a system)
      • Hyperbolic Convex Set
    • Section 22
      • Consequences(of a system)
      • Directed Graph

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• Measure Theory, Paul R. Halmos
  • Chapter 2
    • Set Function
    • Field(Algebra)
    • Sigma-Field(Algebra)
    • Sigma-Finite
    • Totally Sigma-Finite
    • Subtractive
    • Hereditary
  • Chapter 3
    • Borel Set
    • Difference Set
  • Chapter 4
    • Simple Function
    • Essentially Supremum
    • Fundemental(almost everywhere)
    • Fundemental(in measure)
  • Chapter 5
    • Integrable(of simple functions)
    • Indefinite Integral
    • Mean Fundemental(fundemental in the mean)
    • Distance(of integrable functions)
    • Integrable(of finite measurable functions)
  • Chapter 6
    • Positive Set(w.r.t. signed measures)
    • Negative Set(w.r.t. signed measures)
    • Upper Variation(of signed measures)
    • Lower Variation(of signed measures)
    • Total Variation(of signed measures)
    • Bounded Variation(of signed measures)
    • Equivalence Relation(between signed measures)
    • Singular/Mutually Singular
  • Chapter 7

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Theorems

• Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis
  • Chapter 1
    • Theorem 1.1 Helly, Hahn – Banach analytic form
      • Corollary 1.1.1
      • Corollary 1.1.2
      • Corollary 1.1.3
      • Lemma 1.1 Zorn’s Lemma
      • Theorem 1.3 Hahn-Banach, 1st geometric form
        • Lemma 1.2
        • Lemma 1.3
      • Theorem 1.4 Hahn-Banach, 2nd geometric form
        • Corollary 1.4.1
      • Theorem 1.7 Frenchel – Moreau
      • Theorem 1.8 Frenchel – Rockafellar
        • Lemma 1.4
      • Propositions(1.2, 1.5, 1.6)
  • Chapter 2
• Convex Analysis, Rockafellar RT
  • Chapter 1
• Measure Theory, Paul R. Halmos
  • Chapter 2