# Category:Normed Vector Spaces

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

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

Let $\struct {K, +, \circ}$ be a normed division ring.

Let $V$ be a vector space over $K$.

Let $\norm {\,\cdot\,}$ be a norm on $V$.

Then $\struct {V, \norm {\,\cdot\,} }$ is a **normed vector space**.

## Subcategories

This category has the following 27 subcategories, out of 27 total.

### B

- Bounded Normed Vector Spaces (empty)

### C

### E

### L

### M

### N

### O

### Q

- Quotient Norms (1 P)

### R

- Reflexive Spaces (3 P)

### S

- Schauder Bases (5 P)

### W

## Pages in category "Normed Vector Spaces"

The following 54 pages are in this category, out of 54 total.

### C

- Cauchy Sequence is Bounded/Normed Vector Space
- Characteristics of Birkhoff-James Orthogonality
- Characterization of Unit Open Balls of Norms of Euclidean Space
- Characterization of von Neumann-Boundedness in Normed Vector Space
- Closed and Bounded Subset of Normed Vector Space is not necessarily Compact
- Closed Ball is Connected
- Closed Ball is Path-Connected
- Closed Ball is Simply Connected
- Closed Convex Hull in Normed Vector Space is Convex
- Closed Unit Ball in Normed Vector Space is Weakly Closed
- Closure of Convex Subset in Normed Vector Space is Convex
- Compact Subset of Normed Vector Space is Closed and Bounded
- Composite of Continuous Mappings between Normed Vector Spaces is Continuous
- Continuity of Linear Functionals
- Continuity of Linear Transformations/Normed Vector Space
- Convergence of Product of Convergent Scalar Sequence and Convergent Vector Sequence in Normed Vector Space
- Convergent Sequence in Normed Vector Space has Unique Limit

### H

### I

### L

### N

- Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space
- Norm of Continuous Function is Continuous
- Norm of Summation
- Norm on Space of Bounded Linear Transformations is Norm
- Norm satisfying Parallelogram Law induced by Inner Product
- Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact
- Normed Vector Space is Hausdorff
- Normed Vector Space is Hausdorff Locally Convex Space
- Normed Vector Space is Locally Convex Space
- Normed Vector Space is Topological Vector Space
- Normed Vector Space over Complex Numbers with Schauder Basis is Separable
- Normed Vector Space with Schauder Basis is Separable
- Norms on Finite-Dimensional Real Vector Space are Equivalent

### O

### P

### S

- Set Closure is Smallest Closed Set/Normed Vector Space
- Singleton in Normed Vector Space is Closed
- Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition
- Subspace of Normed Vector Space with Induced Norm forms Normed Vector Space