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 19 subcategories, out of 19 total.
Pages in category "Normed Division Rings"
The following 106 pages are in this category, out of 106 total.
C
- Cauchy Sequence Converges Iff Equivalent to Constant Sequence
- Cauchy Sequence in Normed Division Ring is Bounded
- Cauchy Sequence of Subring iff Cauchy Sequence of Normed Division Ring
- 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
- Definition:Center of Sphere in 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
M
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 Division Ring Sequence Converges in Completion iff Sequence Represents Limit
- 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 Sequence Test for Convergence
- Null Sequences form Maximal Left and Right Ideal
O
P
Q
R
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 Cauchy Sequence in Normed Division Ring is 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