Section
Convex Analysis
Convex sets, convex functions, separation theorems, and the Hahn-Banach theorem
Core idea
Click any term to expand its definition inline.
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Metric Spaces
Definitions
- Metric and metric space
- Open and closed balls
- Open set
- Closed set
- Interior
- Closure
- Convergence of a sequence
- Cauchy sequence
- Complete metric space
- Distance function to a set
- Product space
Properties
- Open balls are open
- Closed balls are closed
- Basic properties of open sets
- Basic properties of closed sets
- Basic properties of interior
- Basic properties of closure
- Interior via balls
- Closure via balls
- Closure via sequences
- Closed sets via sequences (I)
- Closed sets via sequences (II)
- Uniqueness of limits
- Convergent sequences are bounded
- Convergent sequences are Cauchy
- Cauchy sequences are bounded
- Cauchy + convergent subsequence implies convergent
- Subsequences converge to the same limit
- Subsequence index bound
- Completeness and closedness
- Completeness of R^k
- Reverse triangle inequality
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Normed Vector Spaces
Definitions
- Norm and normed vector space
- Seminorm
- Bounded sets and sequences
- Convergence in normed spaces
- Bounded linear functional
- Dual space and duality pairing
Properties
- Norm induces a metric
- Convergence implies convergence of norms
- Uniqueness of limits and boundedness
- Algebra of limits
- Existence of a norming functional
- Continuity via closed level sets
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Vector Spaces and Linear Algebra
Definitions
- Linear subspace
- Linear combination
- Span
- Linear independence and dependence
- Basis and dimension
- Direct sum of subspaces
- Linear operator
- Image, kernel, and isomorphism
- Quotient vector space
- Codimension
- Self-adjoint operator
- Positive-semidefinite operator
Properties
- Subspace test
- Intersection of subspaces
- Sum of subspaces equals span of union
- Span equals finite linear combinations
- Extension to a basis
- Existence of a basis
- Bases are maximal linearly independent
- Characterization of direct sums
- Images and preimages of subspaces
- Isomorphism theorem
- Codimension-one subspaces
- Kernel has codimension one
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Convex Sets
Definitions
- Convex set
- Convex combination
- Convex hull
- Line segments
- Set operations (sum, scalar multiple)
- Set-valued mapping
- Balanced and absorbing sets
Properties
- Convexity via convex combinations
- Convex hull is smallest
- Convex hull via combinations
- Intersections are convex
- Sums and scalar multiples are convex
- Cartesian products are convex
- Interior and closure are convex
- Interior-closure relations
- Segments from interior stay in interior
- Closure of intersections
- Operations preserving convexity
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Affine Sets
Definitions
- Affine set
- Affine hull and affine combination
- Affine mapping
- Line connecting two points
- Parallel affine set
Properties
- Properties of affine sets
- Affine sets are translates of subspaces
- Parallel subspace
- Characterization of affine mappings
- Affine images preserve convexity
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Algebraic Interior (Core)
Definitions
Properties
- Core via absorbing translations
- Core is convex
- Core equals interior (normed spaces)
- Segments from core stay in core
- Idempotence of core
- Linear closure is convex
- Linear closure equals topological closure
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Convex Functions
Definitions
- Extended reals and conventions
- Domain, epigraph, proper function
- Convex function via epigraph
- Strictly convex function
- Quasiconvex function
- Indicator function
- Marginal (optimal value) function
Properties
- Equivalent characterizations
- Domain is convex
- Convexity via extension
- Quasiconvexity via sublevel sets
- Slope inequalities
- Convexity via monotone derivative
- Convexity via second derivative
- Convexity via Hessian
- Monotone convex composition
- Affine composition
- Supremum is convex
- Marginal function is convex
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Minkowski Gauge and Sublinear Functions
Definitions
Properties
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Hyperplanes and Separation
Definitions
Separation Theorems
- Separation by a hyperplane
- Separation via sup/inf inequality
- Separation of point and subspace
- Separation via core
- Auxiliary separation lemma
- Separation of two convex sets
- Complex separation (real parts)
- Separation by closed hyperplane
- Closed hyperplane (interior condition)
- Strict separation (open set)
- Strict separation (closed hyperplane)
- Strict separation (compact and closed)
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Hahn-Banach Theorems
- Hahn-Banach (real vector spaces)
- Hahn-Banach (seminorm domination)
- Hahn-Banach (complex vector spaces)
- Hahn-Banach (normed spaces)
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Inequalities
- Nonnegative real below every epsilon is zero
- Weighted AM-GM inequality
- Young's inequality
- Hölder inequality (finite sums)
- Hölder inequality (integrals)
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