# Category:Vector Spaces

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

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

Let $\struct {K, +_K, \times_K}$ be a field.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module.

Then $\struct {G, +_G, \circ}_K$ is a **vector space over $K$** or a **$K$-vector space**.

That is, a **vector space** is a unitary module whose scalar ring is a field.

## Subcategories

This category has the following 41 subcategories, out of 41 total.

### A

### B

- Balanced Sets (16 P)

### C

### D

- Dilation Mappings (2 P)

### E

### F

- Fredholm Operators (3 P)

### G

### H

- Hyperplanes (2 P)

### I

### L

- Linear Independence (12 P)

### M

### N

### Q

- Quasinorms (1 P)

### S

- Scalar Addition (empty)
- Seminorms (15 P)
- Standard Ordered Bases (5 P)

### T

### V

## Pages in category "Vector Spaces"

The following 33 pages are in this category, out of 33 total.

### C

### D

### F

### I

### S

- Set of Vectors defined by Directed Line Segments in Space forms Vector Space
- Singleton is Convex Set
- Size of Linearly Independent Subset is at Most Size of Finite Generator
- Space of Real-Valued Measurable Functions Identified by A.E. Equality is Vector Space
- Star Convex Set is Path-Connected
- Star Convex Set is Simply Connected
- Sum of Union of Subsets of Vector Space and Subset