Banach Space

★★★★☆Undergraduate

📖Definition

Complete normed space. Normed vector space where every Cauchy sequence converges

||x|| ≥ 0, ||x|| = 0 ⟺ x = 0

||αx|| = |α| ||x||

||x + y|| ≤ ||x|| + ||y||

Completeness: Cauchy sequences converge

예제 1

ℝⁿ, Cⁿ (finite dim)

예제 2

Lᵖ(Ω), C([a,b])

예제 3

ℓᵖ (sequence spaces)

Differential equations

Quantum mechanics

Optimization

#바나흐#Banach#완비#normed