Functional Analysis

2.

Normed Vector Spaces

2.1.

Definition and Basic Properties

2.1.1.

Definition of a Norm

2.1.2.

2.1.3.

Examples of Norms

2.1.3.1.

2.1.3.2.

2.1.3.3.

2.1.4.

The Metric Induced by a Norm

2.1.5.

2.2.

Examples of Normed Spaces

2.2.1.

Finite Dimensional Spaces

2.2.1.1.

ℝⁿ with Various Norms

2.2.1.2.

ℂⁿ with Various Norms

2.2.2.

Sequence Spaces

2.2.2.1.

ℓₚ Spaces for 1 ≤ p ≤ ∞

2.2.2.2.

Properties of ℓₚ Spaces

2.2.2.3.

2.2.2.4.

2.2.3.

Function Spaces

2.2.3.1.

C[a,b] with Supremum Norm

2.2.3.2.

C(K) for Compact K

2.2.3.3.

2.2.3.3.1.

Definition via Lebesgue Integration

2.2.3.3.2.

Properties of Lₚ Spaces

2.2.3.3.3.

Hölder's Inequality

2.2.3.3.4.

Minkowski's Inequality

2.3.

Subspaces and Quotients

2.3.1.

Subspaces of Normed Spaces

2.3.2.

Closed Subspaces

2.3.3.

Quotient Spaces

2.3.3.1.

Construction of Quotient Norm

2.3.3.2.

Properties of Quotient Spaces

2.4.

Product and Direct Sum Spaces

2.4.1.

2.4.2.

Direct Sum of Normed Spaces

2.5.

Convergence and Topology

2.5.1.

Norm Convergence

2.5.2.

Cauchy Sequences

2.5.3.

Open and Closed Sets

2.5.4.

Interior, Closure, and Boundary

2.5.5.

2.6.

Continuity and Boundedness

2.6.1.

Continuous Functions

2.6.2.

Uniform Continuity

2.6.3.

2.6.4.

Totally Bounded Sets

2.7.

Equivalence of Norms

2.7.1.

Definition of Equivalent Norms

2.7.2.

Criteria for Norm Equivalence

2.7.3.

Equivalence Classes of Norms

2.8.

Finite Dimensional Normed Spaces

2.8.1.

Equivalence of All Norms

2.8.2.

Compactness Properties

2.8.3.

2.8.4.

Local Compactness

Previous

1. Preliminaries and Foundational Concepts

Go to top

Next

3. Banach Spaces