# Topological Vector Spaces Topics

## 1. Topological Vector Spaces

###### 1.1. General Vector Spaces over $\mathbb{R}$ or $\mathbb{C}$

- Vector Spaces over the Field of Real or Complex Numbers
- Spanning Sets of Vectors
- Linearly Independent Sets of Vectors
- Bases for a Vector Space
- Every Vector Space has a Base

- Convex and Balanced Sets of Vectors
- Properties of Convex Sets of Vectors
- Properties of Balanced Sets of Vectors
- Absolutely Convex Sets of Vectors
- Absorbent Sets of Vectors
- Properties of Absorbent Sets of Vectors

###### 1.2. Topological Spaces

- Topologies and Topological Spaces
- The Interior and Closure of a Set of Points
- Bases of Neighbourhoods for a Point
- The Induced Topology on a Subset of a Topological Space
- Continuous Functions Between Topological Spaces
- Homeomorphisms Between Topological Spaces
- Metric Spaces and Metrizability

###### 1.3. Topological Vector Spaces (TVS) and Locally Convex Topological Vector Spaces (LCTVS) over $\mathbb{R}$ or $\mathbb{C}$

- Topological Vector Spaces over the Field of Real or Complex Numbers
- Bases of Neighbourhoods for a Point in a Topological Vector Space
- The Closure of a Convex Set in a TVS
- The Closure of a Balanced Set in a TVS
- The Closure of an Absolutely Convex Set in a TVS
- Every TVS Has a Base of Closed and Balanced Neighbourhoods of the Origin

- Locally Convex Topological Vector Spaces over the Field of Real or Complex Numbers
- Every LCTVS Has a Base of Closed Absolutely Convex Absorbent Neighbourhoods of the Origin
- Criterion for a Topological Vector Space to be Hausdorff

###### 1.4. Seminorms and Norms

- Seminorms and Norms on Vector Spaces
- The Gauge of an Absolutely Convex and Absorbent Set
- Properties of Gauges of Absolutely Convex and Absorbent Sets
- Continuity of Seminorms on Vector Spaces
- The Coarsest Topology Determined by a Set of Seminorms on a Vector Space
- Criterion for the Coarsest Topology Determined by a Set of Seminorms to be Hausdorff
- Normable Vector Spaces
- Criterion for a LCTVS to be Metrizable

###### 1.5. Examples of Locally Convex Topological Vector Spaces

## Duality

###### 2.1. Linear Operators

###### 2.2. Linear Forms (Functionals)

- Linear Forms on a Vector Space and its Algebraic Dual
- Continuous Linear Forms on a TVS and its Continuous Dual
- Closed Preimage Criterion for a Linear Form to be Continuous in a TVS
- Closures of Subspaces of a Topological Vector Space
- Hyperplanes of a Vector Space

- The Hahn-Banach Theorem for Vector Spaces Part 1
- The Hahn-Banach Theorem for Vector Spaces Part 2
- The Hahn-Banach Theorem for Vector Spaces Part 3
- The Hahn-Banach Theorem for Vector Spaces Part 4
- Corollaries to the Hahn-Banach Theorem for Vector Spaces

###### 2.3. Dual Pairs (E, F) and the Weak Topology σ(E, F)

- Dual Pairs of Vector Spaces
- The Weak Topology on E Determined by F
- The Topological Dual of E Equipped with σ(E, F) is F
- Topologies of the Dual Pair (E, F)

###### 2.4. Polar Sets and Bipolar Sets

###### 2.5. Finite-Dimensional Vector Spaces

###### References

- 1. Topological Vector Spaces (2nd Edition) by A.P. Robertson and Wendy Robertson