# Category:Topological Vector Spaces

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This category contains results about **Topological Vector Spaces**.

Definitions specific to this category can be found in Definitions/Topological Vector Spaces.

Let $\struct {K, +_K, \circ_K, \tau_K}$ be a topological field.

Let $\struct {X, +_X, \circ_X, \tau_X}$ be a vector space over $K$.

Let $\tau_X \times \tau_X$ be the product topology of $\tau_X$ and $\tau_X$.

Let $\tau_K \times \tau_X$ be the product topology of $\tau_K$ and $\tau_X$.

We say that $\struct {X, \tau_X}$ is called a **topological vector space** if and only if:

\((1)\) | $:$ | $+_X: \struct {X \times X, \tau_X \times \tau_X} \to \struct {X, \tau_X}$ is continuous | |||||||

\((2)\) | $:$ | $\circ_X : \struct {K \times X, \tau_K \times \tau_X} \to \struct {X, \tau_X}$ is continuous |

## Subcategories

This category has the following 23 subcategories, out of 23 total.

## Pages in category "Topological Vector Spaces"

The following 67 pages are in this category, out of 67 total.

### B

### C

- Cartesian Space of Topological Field is Topological Vector Space
- Characterization of Continuous Linear Functionals on Topological Vector Space
- Characterization of Continuous Linear Transformation from Metrizable Topological Vector Space to Topological Vector Space
- Choquet's Theorem
- Classification of Open Neighborhoods in Topological Vector Space
- Closure of Balanced Set in Topological Vector Space is Balanced
- Closure of Convex Set in Topological Vector Space is Convex
- Closure of Linear Subspace of Topological Vector Space is Linear Subspace
- Condition for Point being in Closure/Topological Vector Space
- Continuous Real Linear Functional on Complex Topological Vector Space is Real Part of Continuous Complex Linear Functional
- Convergent Sequence in Topological Vector Space is Cauchy
- Convex Open Neighborhood of Origin in Topological Vector Space contains Balanced Convex Open Neighborhood
- Convex Set is Path-Connected
- Convex Subset of Topological Vector Space containing Zero Vector in Interior is Absorbing Set

### D

- Dilation Mapping on Topological Vector Space is Continuous
- Dilation Mapping on Topological Vector Space is Homeomorphism
- Dilation of Closed Set in Topological Vector Space is Closed Set
- Dilation of Closure of Set in Topological Vector Space is Closure of Dilation
- Dilation of Compact Set in Topological Vector Space is Compact
- Dilation of Interior of Set in Topological Vector Space is Interior of Dilation
- Dilation of Open Set in Topological Vector Space is Open
- Direct Product of Topological Vector Spaces is Hausdorff iff Hausdorff Factor Spaces
- Direct Product of Topological Vector Spaces is Topological Vector Space
- Disjoint Compact Set and Closed Set in Topological Vector Space separated by Open Neighborhood

### F

### I

- Interior of Balanced Set containing Origin in Topological Vector Space is Balanced
- Interior of Convex Set in Topological Vector Space is Convex
- Interior of Translation of Set in Topological Vector Space is Translation of Interior
- Isomorphism from Cartesian Space to Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Homeomorphism

### L

- Linear Combination of Continuous Functions valued in Topological Vector Space is Continuous
- Linear Combination of Convergent Sequences in Topological Vector Space is Convergent
- Linear Transformation between Topological Vector Spaces Continuous iff Continuous at Origin
- Linear Transformation between Topological Vector Spaces is Open iff Interior of Image of Open Neighborhood contains Zero Vector
- Linear Transformation from Cartesian Space on Hausdorff Topological Field to Topological Vector Space is Continuous
- Locally Convex Space is Topological Vector Space

### N

### O

- Open Neighborhood of Dilation of Point in Topological Vector Space contains Pointwise Scalar Multiplication of Open Neighborhood of Scalar with Open Neighborhood of Vector
- Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood
- Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods

### P

### S

- Scalar Multiple of Convergent Net in Topological Vector Space is Convergent
- Scalar Multiple of Convergent Sequence in Topological Vector Space is Convergent
- Sum of Closures is Subset of Closure of Sum in Topological Vector Space
- Sum of Compact Subsets of Topological Vector Space is Compact
- Sum of Convergent Nets in Topological Vector Space is Convergent
- Sum of Convergent Sequences in Topological Vector Space is Convergent
- Sum of Set and Open Set in Topological Vector Space is Open

### T

- Topological Vector Space as Union of Dilations of Open Neighborhood of Origin
- Topological Vector Space is Hausdorff iff T1
- Topological Vector Space over Connected Topological Field is Connected
- Translation Mapping on Topological Vector Space is Continuous
- Translation Mapping on Topological Vector Space is Homeomorphism
- Translation of Closed Set in Topological Vector Space is Closed Set
- Translation of Local Basis in Topological Vector Space
- Translation of Open Set in Topological Vector Space is Open