Semester 4 / Linear Algebra /Hilbert And Banach Space

Hilbert & Banach Space

1 min read Updated Fri Apr 24 2026 03:19:45 GMT+0000 (Coordinated Universal Time)
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Cauchy sequences can be defined on metric spaces.

Cauchy Sequence

A sequence u:Z+→Au:\mathbb{Z}^+ \rightarrow A is Cauchy iff:

∀ϵ>0 ∃N∈Z+ ∀m,n;m,n>N  ⟹  ∣un−um∣<ϵ\forall \epsilon \gt 0\, \exists N \in \mathbb{Z}^+\, \forall m,n; m,n \gt N \implies \lvert u_n - u_m \rvert \lt \epsilon

Complete

A set is said to be complete iff all Cauchy sequences approach a limit in the set.

Hilbert Space

A complete inner product space.

Banach Space

A complete normed space.