# 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
- Review of General Vector Spaces

###### 1.2. Convex, Balanced, Absolutely Convex, and Absorbent Sets

- 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
- Review of Convex, Balanced, Absolutely Convex, and Absorbent Sets

###### 1.3. 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
- Review of Topological Spaces

###### 1.4. 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
- Criterion for a Topological Vector Space to be Hausdorff
- 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
- Review of Topological Vector Spaces

###### 1.5. 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
- Review of Seminorms and Norms

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

- Examples of LCTVS - Spaces of Continuous Real-Valued or Complex-Valued Functions
- Examples of LCTVS - Spaces of Infinitely Differentiable Functions
- Examples of LCTVS - Sequence Spaces

###### 1.7. Full Review of Chapter 1

## 2. Duality

###### 2.1. Linear Operators

- Linear Operators Between Vector Spaces
- Isomorphic Topological Vector Spaces
- Review of 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
- The Hahn-Banach Separation Theorem
- Review of Linear Forms and the Hahn-Banach Theorem

###### 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)
- Review of Dual Pairs

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

- The Polar of a Set
- The Polar of a Subspace
- The Polar Criterion for Equicontinuity of a Set of Linear Forms
- The Bipolar of a Set
- If E ⊆ G ⊆ F* and (E, F) is a Dual Pair, A ⊆ A°°
- If E ⊆ G ⊆ F* and (E, F) is a Dual Pair then A°° is the σ(G, F)-Closed Absolutely Convex Hull of A
- If E is a Hausdorff LCTVS then A°° (in E) is the Closed Absolutely Convex Hull of A
- The Polar of an Intersection of σ(E, F)-Closed Absolutely Convex Sets
- Review of Polar and Bipolar Sets

###### 2.5. The Transpose of a Linear Operator

- The Transpose of a Linear Operator
- Weakly Continuous Linear Operators
- For Hausdorff LCTVS E and F, if t from E to F is Continuous then t is Weakly Continuous
- For Dual Pairs (E, F), (G, H), and a Weakly Continuous Linear Operator t from E to G, (t(A))° = t'-1(A°)
- Review of Tranposes

###### 2.6. Finite-Dimensional Vector Spaces

- The Dual Base for E* of a Finite-Dimensional Vector Space E
- Finite-Dimensional Vector Spaces Have a Unique Locally Convex and Hausdorff Topology
- Finite-Dimensional Subspaces are Closed in a Hausdorff LCTVS
- Review of Finite-Dimensional Vector Spaces

###### 2.7. Full Review of Chapter 2

## 3. Topologies on Dual Spaces

###### 3.1. Bounded Sets in a Locally Convex Topological Vector Space

- Bounded Sets in a LCTVS
- The Closure, Convex Hull, and Absolutely Convex Hull of a Bounded Set is a Bounded Set in a LCTVS
- Subsets, Scalar Multiples, Finite Unions, and Arbitrary Intersections of Bounded Sets in a LCTVS
- A Continuous Linear Image of a Bounded Set is a Bounded Set in a LCTVS
- Bounded Neighbourhood Criterion for a Hausdorff LCTVS to be Normable
- Review of Bounded Sets in a LCTVS

###### 3.2. Polar Topologies

- Criteria for a Subset A to be σ(E, F)-Weakly Bounded
- Polar Topologies
- Equicontinuity Classification of Hausdorff Locally Compact Topologies
- Weak Continuity Criterion of t for the Continuity of t'
- For Normed Spaces E and F, t is Continuous IFF t' is Continuous
- Review of Polar Topologies

###### 3.3. Precompact Sets and Compact Sets

- Precompact Sets in a LCTVS
- Basic Properties of Precompact Sets in a LCTVS
- Boundedness of Precompact Sets in a LCTVS
- For Hausdorff LCTVS, Precompact Neighbourhood Implies Finite-Dimensional
- Results Regarding Precompactness
- Compact Sets in a Topological Space
- Basic Properties of Compact Sets in a LCTVS
- Review of Precompact Sets and Compact Sets

###### 3.4. Filters and Ultrafilters

- Filters and Filter Bases
- Convergence of Filters and Filter Bases in a Topological Space
- The Elementary Filter Associated with a Sequence
- Ultrafilters
- Review of Filters and Ultrafilters

###### 3.5. Completeness

###### References

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