The Field Axioms of ℝ Notes

1. History of analysis

Real analysis has a very wide history that spans millennia.

1.1. Modern analysis

1. Set Theory and Pathological Examples

• Cantor developed set theory and introduced the concept of cardinality and uncountability (e.g., ℝ is uncountable).

• Weierstrass and others constructed nowhere-differentiable continuous functions, showing that intuition could fail without rigor.

2. Measure Theory and Lebesgue Integration

• Lebesgue introduced a new way to define integration, based on measurable sets and functions, solving many of Riemann’s shortcomings.

• This became the foundation of modern probability theory and functional analysis.

3. Real Analysis Today

• Real analysis now includes:

  • Topology (open sets, compactness, connectedness)

  • Metric and normed spaces

  • Measure and integration theory

  • Applications in PDEs, ergodic theory, stochastic processes, and modern physics

1.2. The Age of Rigor and the Real Numbers (1800s)

1. The Demand for Precision

• Cauchy introduced limits, continuity, and convergent sequences using informal definitions.

• Weierstrass formalized these ideas using precise ε-δ (epsilon-delta) arguments, eliminating reliance on intuition.

2. Constructing the Real Line

• Mathematicians realized that a rigorous understanding of the real numbers was essential.

• Dedekind introduced Dedekind cuts to construct real numbers from rationals.

• Cantor used Cauchy sequences to complete ℚ into ℝ, formalizing the notion of completeness.

3. Early Integration Theory

• Riemann defined integration via finite sums over partitions, forming the Riemann integral.

• This worked well for many functions but had limitations—especially for discontinuous or unbounded functions.

1.3. Foundations and the Rise of Calculus (Ancient to 1700s)

1. Greek and Pre-Modern Roots

• Eudoxus developed the theory of proportions, avoiding irrational numbers by working with ratios.

• Archimedes used the method of exhaustion to approximate areas and volumes—an early use of limits.

2. Birth of Calculus

• In the 17th century, Newton and Leibniz independently invented calculus.

• They relied on intuitive ideas like infinitesimals and fluxions—which lacked rigorous definitions.

• Calculus allowed for major advances in physics and engineering, but its logical foundation was unclear.

2. Field Axioms

We will be discussing the different field axioms in R

2.1. Distributive law and Field Structure

This axiom connects addition and multiplication:

1. Distributivity of Multiplication over Addition:

   $a(b + c) = ab + ac$

   a(b+c)=ab+ac

2.2. Axioms of Multiplication

These axioms govern multiplication in ℝ:

1. Associativity of Multiplication:

   $a \cdot (b \cdot c) = (a \cdot b) \cdot c$

   a⋅(b⋅c)=(a⋅b)⋅c

2. Commutativity of Multiplication:

   $ab = ba$

   ab=ba

3. Multiplicative Identity:
   There exists

   $1 \in ℝ$

   1∈R,

   $1 \ne 0$

   1=0, such that

   $a \cdot 1 = a$

   a⋅1=a

4. Multiplicative Inverses:
   For each

   $a \ne 0$

   a=0, there exists

   $a^{-1} \in ℝ$

   a−1∈R such that

   $a \cdot a^{-1} = 1$

   a⋅a−1=1

These axioms make

$ℝ \setminus \{0\}$

R∖{0} into an abelian group under multiplication.

2.3. Axioms of addition

These axioms define the structure of addition in the real numbers:

1. Associativity of Addition:

   $a + (b + c) = (a + b) + c$

   a+(b+c)=(a+b)+c

2. Commutativity of Addition:

   $a + b = b + a$

   a+b=b+a

3. Additive Identity:
   There exists a

   $0 \in ℝ$

   0∈R such that

   $a + 0 = a$

   a+0=a

4. Additive Inverses:
   For each

   $a \in ℝ$

   a∈R, there exists

   $-a \in ℝ$

   −a∈R such that

   $a + (-a) = 0$

   a+(−a)=0

These form the abelian group structure of

$(ℝ, +)$

(R,+).

3. Vector Spaces

In this chapter, we will be investigating properties of vectors and vector spaces

3.1. Examples and Non-Examples

This subsection grounds the abstract definition in concrete examples.

Examples of Vector Spaces:

• $ℝ^n$

  Rn: Standard Euclidean space (e.g.,

  $ℝ^2$

  R2,

  $ℝ^3$

  R3)

• The space of all polynomials

  $\mathbb{P}$

  P

• The set of all real-valued continuous functions on an interval

• $M_{m \times n}(ℝ)$

  Mm×n​(R): All real

  $m \times n$

  m×n matrices

Non-Examples:

• $ℕ$

  N under normal addition and scalar multiplication (no additive inverses)

• Any set where scalar multiplication is not defined over a field

Emphasize why these are or aren't vector spaces—this develops students' intuition.

3.2. Definition and Axioms of Vector Spaces

This subsection formalizes what it means for a set to be a vector space over a field (like ℝ).

Definition:
A vector space

V over a field

F is a set equipped with:

• Vector addition:

  $+ : V \times V \to V$

  +:V×V→V

• Scalar multiplication:

  $\cdot : F \times V \to V$

  ⋅:F×V→V
  that satisfy the following 8 axioms for all

  $u, v, w \in V$

  u,v,w∈V and

  $a, b \in F$

  a,b∈F:

1. $u + v = v + u$

   u+v=v+u (commutativity)

2. $(u + v) + w = u + (v + w)$

   (u+v)+w=u+(v+w) (associativity of addition)

3. There exists a zero vector

   $0 \in V$

   0∈V such that

   $v + 0 = v$

   v+0=v

4. Each

   $v \in V$

   v∈V has an additive inverse

   $-v \in V$

   −v∈V such that

   $v + (-v) = 0$

   v+(−v)=0

5. $a(u + v) = au + av$

   a(u+v)=au+av (distributivity over vector addition)

6. $(a + b)v = av + bv$

   (a+b)v=av+bv (distributivity over field addition)

7. $a(bv) = (ab)v$

   a(bv)=(ab)v (compatibility of scalar multiplication)

8. $1 \cdot v = v$

   1⋅v=v (identity element of scalar multiplication)

3.3. ubspaces, Linear Combinations, and Span

This subsection introduces the structure within vector spaces.

Subspaces:
A subspace

W⊆V is a subset that is itself a vector space under the same operations.

Linear Combination:
Any expression of the form

a1​v1​+a2​v2​+⋯+an​vn​, where

ai​∈F and

vi​∈V

Span:
The set of all linear combinations of a set of vectors. Denoted

span{v1​,...,vn​}

Subspaces and spans form the gateway to understanding bases and dimension, key ideas in both linear algebra and analysis.