Course Introduction and Motivation

• Course: Introduction to Functional Analysis (first lecture)
• Prerequisites: Linear algebra, calculus, real analysis (1800b)
• Core problem: Previous courses handle finitely many independent variables (finite dimensions)
  • Calculus: functions of 1-4 variables (R, R², R³, R⁴)
  • Linear algebra: systems with finite equations/unknowns
• Need for functional analysis: Solve problems with infinitely many independent variables
  • Examples: ODEs, PDEs, optimization problems
  • Independent variables become functions (infinite dimensional vector spaces)
  • Example: shortest curve between two points requires infinitely many numbers to specify

Vector Spaces Review

• Definition: Vector space V over field K (R or C)
• Operations:
  • Addition: V × V → V (v₁ + v₂)
  • Scalar multiplication: K × V → V (αv)
• Examples:
  • Finite dimensional: R², Rⁿ, Cⁿ
  • Infinite dimensional: C[0,1] (continuous functions on [0,1])

Finite vs Infinite Dimensional Spaces

• Finite dimensional: Every linearly independent set is finite
  • Linear independence: α₁v₁ + ... + αₙvₙ = 0 implies all αᵢ = 0
• Infinite dimensional: Contains infinite linearly independent sets
• Example: C[0,1] is infinite dimensional
  • Set {fₙ(x) = xⁿ : n ∈ ℕ ∪ {0}} is linearly independent
• Key challenge: Heine-Borel theorem fails in infinite dimensions
  • Closed and bounded sets not necessarily compact
  • Need new machinery for analysis

Norms and Normed Spaces

• Norm definition: Function ‖·‖: V → [0,∞) satisfying:
  1. Definiteness: ‖v‖ = 0 ⟺ v = 0
  2. Homogeneity: ‖λv‖ = |λ|‖v‖ for all λ ∈ K, v ∈ V
  3. Triangle inequality: ‖v₁ + v₂‖ ≤ ‖v₁‖ + ‖v₂‖
• Seminorm: Satisfies homogeneity and triangle inequality but not definiteness
• Normed space: Vector space equipped with norm

Induced Metrics

• Metric from norm: d(v,w) = ‖v - w‖
• Metric properties:
  • d(x,y) = 0 ⟺ x = y
  • d(x,y) = d(y,x) (symmetry)
  • d(x,z) ≤ d(x,y) + d(y,z) (triangle inequality)
• Norm properties 1,2,3 imply metric properties a,b,c respectively

Examples of Norms

Finite Dimensional Spaces (Rⁿ, Cⁿ)

• Euclidean norm: ‖x‖₂ = (∑|xᵢ|²)^(1/2)
• ∞-norm: ‖x‖∞ = max₁≤ᵢ≤ₙ |xᵢ|
• Lᵖ-norm: ‖x‖ₚ = (∑|xᵢ|ᵖ)^(1/p) for 1 ≤ p < ∞
• Unit balls visualization in R²:
  • L² ball: circle
  • L∞ ball: square
  • L¹ ball: diamond (tilted square)
  • Other Lᵖ balls: between L¹ and L∞

Function Spaces

• Space: C∞(X) = bounded continuous functions on metric space X
• Uniform norm: ‖f‖∞ = sup_{x∈X} |f(x)|
• Triangle inequality proof: |u(x) + v(x)| ≤ |u(x)| + |v(x)| ≤ ‖u‖∞ + ‖v‖∞
• Convergence: uₙ → u in uniform norm ⟺ uniform convergence on X

Sequence Spaces

• ℓᵖ spaces: Sequences a = (aⱼ) with finite Lᵖ norm
• ℓᵖ norm: ‖a‖ₚ = (∑|aⱼ|ᵖ)^(1/p) for 1 ≤ p < ∞
• ℓ∞ norm: ‖a‖∞ = sup_j |aⱼ|
• Example: Sequence (1/j) ∈ ℓᵖ for p > 1 but not for p = 1

Banach Spaces

• Definition: Normed space that is complete with respect to induced metric
• Completeness: Every Cauchy sequence converges in the space
• Named after: Stefan Banach

Examples of Banach Spaces

• Finite dimensional: Rⁿ, Cⁿ with any Lᵖ norm (complete)
• Sequence spaces: ℓᵖ for 1 ≤ p ≤ ∞ (complete)
• Function spaces: C∞(X) for complete metric space X

Completeness Proof for C∞(X)

Theorem

If X is complete metric space, then C∞(X) is Banach space

Proof Strategy

1. Boundedness: Cauchy sequence {uₙ} is bounded in C∞(X)
2. Pointwise convergence: For each x ∈ X, {uₙ(x)} is Cauchy in C, hence converges
3. Candidate function: u(x) = lim_{n→∞} uₙ(x)
4. Uniform convergence: Show ‖uₙ - u‖∞ → 0
5. Continuity: u is continuous as uniform limit of continuous functions

Key Steps

• Cauchy condition: ∀ε > 0, ∃N: ∀n,m ≥ N, ‖uₙ - uₘ‖∞ < ε/2
• Pointwise bound: |uₙ(x) - uₘ(x)| ≤ ‖uₙ - uₘ‖∞ < ε/2
• Taking limit: |uₙ(x) - u(x)| ≤ ε/2 for all x ∈ X
• Uniform bound: ‖uₙ - u‖∞ ≤ ε/2 < ε

Additional Banach Space Examples

• c₀ space: Sequences in ℓ∞ with lim_{j→∞} aⱼ = 0
• Norm: ‖a‖∞ = sup_j |aⱼ|
• Subspace: c₀ ⊂ ℓ∞ (closed subspace of Banach space)