Banach Space

Normed Spaces and Banach Spaces are discussed here.

Normed Spaces

• Definition of Normed Space
  • Semi-norm
• Bounded Subsets
• Convergence of Sequences
• Norm is a Continuous Mapping
• Uniform Convexity
• Dense Subsets and Closure
• Isometric Mapping
• Embedding, Inclusion Map
  • Every Isometric Mapping is an Embedding
  • Natural Embedding and Reflexive Spaces
• Proof of Riesz’s Lemma
• Proof of Riesz’s Theorem

Finite Dimension

• Hamel Basis of Finite Dimensional Vector Spaces
• Finite Dimensional Normed Spaces Have a Basis
• All Norms on a Finite Dimensional Vector Space are Equivalent
• Finite Dimensional Normed Spaces are Complete

Series

• Series, Span, Total Sequences
• Schauder Basis of Infinite Dimensional Vector Spaces

Operators

• Definition of Operators
  • Integral Operators
• Properties of Linear Operators
• Hahn-Banach Extension Theorem
  • Hahn-Banach Theorem for Real, Complex, Semi-normed Spaces

Bounded Linear Operators

• Properties
• Extension Theorem

Compact Operators

• Compact Operators
• Compactness Conditions

Banach Spaces

• Definition of Banach Space
• Fréchet Derivative
  • Chain Rule
• Proof of Banach Fixed Point Theorem

$\ell^{p}$ Spaces

• $\ell^{p}$ Spaces
• As $p \to \infty$, the $p$-norm becomes the maximum norm

References

• Erwin Kreyszig, Introductory Functional Analysis with Applications (1989)
• Robert A. Adams and John J. F. Foutnier, Sobolev Space (2nd Edition, 2003)
• Ole Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (2010)