Category:Functional Analysis

1

• UTLOC:1

A

• A function is continuous if and only if the pre-image of every basis element is open
• A monotonically increasing sequence bounded above converges
• 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
• Absolute value (object)
• Algebra (linear algebra)

B

• Bernstein polynomial
• Bernstein polynomial/Definition
• Borel sigma-algebra of the real line
• Bounded
• Bounded (linear map)
• Bounded (sequence)
• Bounded linear map
• Bounded sequence
• Bounded set

C

• C( )0,1( ,R) is not complete when considered with the L^1 norm
• C(I,X)
• C(X,Y)
• Cauchy criterion for convergence
• Cauchy sequence
• Cauchy sequence/Definition
• Cauchy sequence/Short definition
• Cauchy-Schwarz inequality for inner product spaces
• Class of smooth real-valued functions on R-n
• Closed interval
• Closure (topology)
• Comparison test for real series
• Comparison test for real series/Statement
• Complex functionals
• Convergence (sequence)
• Convergence of a sequence
• Convergent (sequence)
• Convergent sequence
• Convex
• Convex function
• Convex hull
• Convex set

D

• Dense
• Dense set
• Differentiation
• Distance from a point to a set

E

• Epigraph
• Equivalence of Cauchy sequences

E cont.

• Equivalence of Cauchy sequences/Definition
• 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 set being bounded
• Equivalent conditions to a set being bounded/Statement
• Euclidean n-space
• Every convergent sequence is Cauchy
• Every lingering sequence has a convergent subsequence
• Every sequence in a compact space is a lingering sequence

F

• 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
• Fréchet derivative

G

• Gateaux derivative
• 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

H

• Hilbert space

I

• If a real series converges then its terms tend to zero
• If a subsequence of a Cauchy sequence converges then the Cauchy sequence itself also converges
• If an inner product is non-zero then both arguments are non-zero
• Inequalities for inner products
• Inner product
• Inner product examples
• Inner product space
• Inner-product space
• Interior
• Interior (topology)
• Interior point (topology)
• Intermediate value theorem

L

• Line
• Linear combination
• Lingering sequence
• List of topological properties

M

• Metric induced by a norm
• Metric induced by the norm
• Metrically bounded set
• Metrics
• 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

• Norm
• Normed space
• Normed spaces

N cont.

• Null sequence

O

• Omega(X,b)
• Open ball
• Open neighbourhoods
• Open set
• Operations on convergent sequences of real numbers
• Orthogonal complement
• Orthogonal vectors
• Orthonormal set

P

• Perpendicular

R

• Real line
• Real numbers
• Real sequence
• Real sequences
• Real-valued function
• Reverse triangle inequality

S

• Sequence
• Series (summation)
• Set of all linear maps between spaces
• Set of continuous functions
• Space of square-summable sequences
• Strong derivative
• Sub-sequence
• Subsequence
• Subsequence/Definition

T

• The closure of a linear subspace of a normed space is a linear subspace
• The ell p spaces
• The interior of a set in a topological space is equal to the union of all interior points of that set
• The little-L spaces are complete
• The little-L(C) spaces are complete
• The norm of a space is a uniformly continuous map with respect to the topology it induces
• The real line
• The reals
• The set of continuous functions between topological spaces
• The space of all linear maps between spaces
• The vector space of all linear maps between two spaces
• Topological space induced by the metric space
• Topological space/Definition
• Topology induced by the metric

U

• Unbounded
• Uniform continuity
• Uniformly continuous

W

• Weak derivative
• Weierstrass approximation theorem