# Category:Normed Division Rings

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This category contains results about Normed Division Rings.

Definitions specific to this category can be found in Definitions/Normed Division Rings.

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

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

Then $\struct{R, \norm{\,\cdot\,} }$ is a **normed division ring**.

## Subcategories

This category has the following 15 subcategories, out of 15 total.

### C

### N

### O

### P

## Pages in category "Normed Division Rings"

The following 98 pages are in this category, out of 98 total.

### C

- Cauchy Sequence Converges Iff Equivalent to Constant Sequence
- Cauchy Sequence in Normed Division Ring is Bounded
- Cauchy Sequences form Ring with Unity
- Center is Element of Closed Ball in Normed Division Ring
- Center is Element of Closed Ball/Normed Division Ring
- Center is Element of Open Ball in Normed Division Ring
- Center is Element of Open Ball/Normed Division Ring
- Centers of Closed Balls in Non-Archimedean Division Rings
- Centers of Open Balls in Non-Archimedean Division Rings
- Characterisation of Cauchy Sequence in Non-Archimedean Norm
- Characterisation of Non-Archimedean Division Ring Norms
- Combination Theorem for Cauchy Sequences
- Combination Theorem for Sequences in Normed Division Rings
- Completion of Normed Division Ring
- Constant Sequence Converges to Constant in Normed Division Ring
- Convergent Sequence in Normed Division Ring is Bounded
- Convergent Sequence in Normed Division Ring is Cauchy Sequence
- Convergent Sequence with Finite Elements Prepended is Convergent Sequence
- Convergent Subsequence of Cauchy Sequence in Normed Division Ring

### E

- Embedding Division Ring into Quotient Ring of Cauchy Sequences
- Embedding Normed Division Ring into Ring of Cauchy Sequences
- Equivalence of Definitions of Convergence in Normed Division Rings
- Equivalence of Definitions of Equivalent Division Ring Norms
- Equivalent Cauchy Sequences have Equal Limits of Norm Sequences
- Equivalent Norms are both Non-Archimedean or both Archimedean
- Equivalent Norms on Rational Numbers
- Equivalent Norms on Rational Numbers/Necessary Condition
- Equivalent Norms on Rational Numbers/Sufficient Condition
- Eventually Constant Sequence Converges to Constant

### I

### L

### N

- No Non-Trivial Norm on Rational Numbers is Complete
- Non-Archimedean Division Ring Iff Non-Archimedean Completion
- Non-Archimedean Division Ring is Totally Disconnected
- Non-Archimedean Norm iff Non-Archimedean Metric
- Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition
- Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition
- Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary
- Norm is Complete Iff Equivalent Norm is Complete
- Norm of Difference in Division Ring
- Norm of Integer of Division Ring
- Norm of Inverse in Division Ring
- Norm of Negative in Division Ring
- Norm of Negative of Unity of Division Ring
- Norm of Power Equals Unity in Division Ring
- Norm of Quotient in Division Ring
- Norm of Ring Negative
- Norm of Unity of Division Ring
- Norm Sequence of Cauchy Sequence has Limit
- Normed Division Ring Completions are Isometric and Isomorphic
- Normed Division Ring is Dense Subring of Completion
- Normed Division Ring is Field iff Completion is Field
- Normed Division Ring Operations are Continuous
- Normed Division Ring Operations are Continuous/Addition
- Normed Division Ring Operations are Continuous/Corollary
- Normed Division Ring Operations are Continuous/Inversion
- Normed Division Ring Operations are Continuous/Multiplication
- Normed Division Ring Operations are Continuous/Negation
- Normed Vector Space Requires Multiplicative Norm on Division Ring
- Norms Equivalent to Absolute Value on Rational Numbers
- Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition
- Norms Equivalent to Absolute Value on Rational Numbers/Sufficient Condition
- Null Sequences form Maximal Left and Right Ideal

### O

### P

### Q

### S

- Sequence Converges to Within Half Limit/Normed Division Ring
- Sequence is Bounded in Norm iff Bounded in Metric
- Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition
- Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition
- Sequence of Powers of Number less than One/Normed Division Ring
- Sphere is Set Difference of Closed and Open Ball in Normed Division Ring
- Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring
- Subring of Non-Archimedean Division Ring
- Subsequence is Equivalent to Cauchy Sequence
- Subsequence of a Cauchy Sequence is a Cauchy Sequence

### T

- Three Points in Ultrametric Space have Two Equal Distances/Corollary 2
- Three Points in Ultrametric Space have Two Equal Distances/Corollary 3
- Three Points in Ultrametric Space have Two Equal Distances/Corollary 4
- Three Points in Ultrametric Space have Two Equal Distances/Corollary 5
- Topological Properties of Non-Archimedean Division Rings