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Real Numbers

Last updated Nov 1, 2022

# Definition 1

The unique Complete totally ordered Field. It is usually denoted $\mathbb{R}$ and is used to represent the continuum.

# Remarks

One approach is to axiomatically assume the existence of the Real Numbers
An alternative approach is to construct the reals.

# As an Inner Product Space

Vector Subspace of $\mathbb{C}$.

# Other Outlinks

Complex Numbers

Backlinks

A Real Vector Space over the Rational Field is Infinite-Dimensional
A Sequence Converges on the Extended Reals iff the number of Complete Upcrossings is Finite
A Set is Open in the Metric Topology iff it contains a Ball around each Point
Absolute Convergence
Affine Combination
Affine Set
Cauchy Sequence
Characteristic Function
Closed Halfspace
Closed Line Segment
Upcrossings
Convergence Almost Surely does not imply convergence of moments
Convergence of a Bounded Sequence is determined by Convergent Subsequences
Convergent Sequences are Cauchy
Convex Combination
Convex Function
Convex Set
Correspondence of Lebesgue-Stieltjes Measures and Distribution Functions
Differential Entropy
Differential Entropy of Uniform Random Variable
Epsilon Principle
Euclidean Space is Second Countable
Every Sequence on the Reals contains a Monotone Subsequence
Expectation does not always exist
Extreme Value Theorem
Function Limit
Hyperplane
Indicator Functions on Measureable Sets are in L+
Inner Product
Integer Linear Program
Integration is Non-Decreasing
Kolmogorov Three Series Theorem
Lebesgue-Stieltjes Measure
Left-Continuous
Left Continuous Inverse
Left Function Limit
Line
Martingale Difference Sequence
Monotone Sequences converge to their Extremum
Open Ball
Open Halfspace
Order Topology coincides with Metric Topology on the Real Numbers
Pathwise Connected
Path-Connected implies Connected
Positive Semidefinite Matrices form a Convex Cone
Positive Semidefinite Matrix
Pseudo-Boolean Function
Quotient Norm
Real Numbers
Right Continuous Inverse
Right Function Limit
Scheffe's Lemma
Sigma Algebra induced by an Indicator Function
Solution set of Linear Equations is an Affine Set
Supremum
The Induced Sigma Algebra of a Borel Function is Finite iff it is Simple
Total Orderings that cannot be Order Embedded into the Reals

Real Numbers

# Definition 1

# Remarks

# As an Inner Product Space

# Other Outlinks

Backlinks

Interactive Graph