Category:Analysis

1

• UTLOC:1

A

• 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 monotonically increasing sequence bounded above converges
• 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
• Abs (function)
• Absolute value (function)

B

• Bernstein polynomial
• Bernstein polynomial/Definition
• Borel sigma-algebra of the real line
• Bounded (sequence)
• 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-Schwarz inequality for inner product spaces
• Closed interval
• Comparison test for real series
• Comparison test for real series/Statement
• Complex functionals
• Connected topological space
• Convergent sequence
• Converges (sequence)
• Convex
• Convex function
• Convex hull
• Convex set

D

• Derivative
• Derivative (analysis)
• Differentiation (subject)
• Distance from a point to a set

E

• Epigraph

E cont.

• Epsilon form of inequalities
• Equivalence of Cauchy sequences
• Equivalence of Cauchy sequences/Definition
• Equivalence of Cauchy sequences/Proof
• Equivalent conditions to a set being bounded
• Equivalent conditions to a set being bounded/Statement

F

• Floor (function)
• Floor function
• 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
• Geometric progression
• Geometric sequence
• 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
• Graph

I

• If a real series converges then its terms tend to zero
• If an inner product is non-zero then both arguments are non-zero
• Inner-product space
• Integral
• Intermediate value theorem

K

• Kronecker delta

L

• Line

M

• Measurable functions
• Metrically bounded set
• Metrics
• Minimum
• Minimum function
• Modulo operation
• Modulo operator
• Modulus operation
• 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

• Normed spaces
• Null sequence

O

• Open neighbourhoods
• Operations on convergent sequences of real numbers
• Orthogonal complement
• Orthogonal vectors

R

• Real line
• Real numbers
• Real sequence
• Real sequences

S

• Series
• Series (summation)
• Set of continuous functions
• Notes:Simple function approximation to a numerical function
• Square root
• Square root (real function)
• Strong derivative

T

• The closure of a linear subspace of a normed space is a linear subspace
• The ell p spaces
• 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
• Topology induced by the metric
• Total derivative

U

• Unbounded
• Uniform continuity
• Uniformly continuous
• Usual topology of the reals

W

• Weak derivative
• Weierstrass approximation theorem