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Metric & Normed Spaces - Coggle Diagram
Metric & Normed Spaces
PROPERTIES
Basic Properties: reverse triangle inequality, continuity of metric, diameter, properties of norm-induced metric spaces not pertinent to all metric spaces
Equivalence relations on metric spaces: Homeomorphic, strongly-equivalent and isometric metric spaces, equivalent norms
Isometry
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Specific examples of isometry: any two open intervals (a,b), (c,d) of same length in R
Homeomorphism
General and specific examples of homeomorphism: Isometry, strong-equivalence, any open interval (a,b) and R
Examples of non-homeomorphism: (a,b) and [a,b] in R, R and discrete space, field spaces of different dim.
Strong-Equivalence:
Specific examples of strong-equivalence: any two open intervals (a,b) and (c,d) in R
General examples of equivalent norms: the equivalence of all norms on finite-dim V, Isometry
Completeness
EXAMPLES
General Examples:
- Banach Space, Hilbert space
- Dense subspace condition
- Strong-equivalence criterion
- Closedness criterion (for subspaces)
- Completion = Closure (for subspaces),
- Completions (existence & uniqueness of completion)
Specific Examples:
- Normed field space
- compact MS
- C[0,1] with infinity norm
- C^k[0,1] with C^k norm
Counter-examples:
- Open intervals in R
- Space of rational numbers Q
- Space of zeroing sequences with p-norm
- C^1[0,1] with infinity norm
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