User:IssaRice/Analogous results between subfields of math: Difference between revisions
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==Finiteness vs compactness== | |||
see https://www.math.ucla.edu/~tao/preprints/compactness.pdf | |||
==considering domain of maps== | |||
If A is a set such that every continuous f: A -> R attains a max, then A is compact (?) (found in Schramm's analysis text). | |||
This reminds me of the homomorphism characterization of normal subgroups. A subgroup H is normal iff there is ''some'' f : H -> G such that ker f = H. | |||
There's also a similar characterization for connected sets, i.e. A is connected iff every continuous function f: A->R has the intermediate value property, which says that if f(a) < 0 and f(b) > 0 then there is c in [a,b] such that f(c)=0. (this only works on the real line?) | |||
==Injective iff surjective== | |||
* If A and B are finite sets that have the same cardinality, then <math>f: A \to B</math> is injective iff surjective. | |||
* If V and W are finite-dimensional vector spaces with the same dimension, then a linear map <math>T : V \to W</math> is injective iff surjective. | |||
==See also== | ==See also== | ||
* https://machinelearning.subwiki.org/wiki/User:IssaRice/Subfield_of_math_as_study_of_concepts_preserved_under_transformation | * https://machinelearning.subwiki.org/wiki/User:IssaRice/Subfield_of_math_as_study_of_concepts_preserved_under_transformation | ||
Latest revision as of 20:46, 29 August 2020
Finiteness vs compactness
see https://www.math.ucla.edu/~tao/preprints/compactness.pdf
considering domain of maps
If A is a set such that every continuous f: A -> R attains a max, then A is compact (?) (found in Schramm's analysis text).
This reminds me of the homomorphism characterization of normal subgroups. A subgroup H is normal iff there is some f : H -> G such that ker f = H.
There's also a similar characterization for connected sets, i.e. A is connected iff every continuous function f: A->R has the intermediate value property, which says that if f(a) < 0 and f(b) > 0 then there is c in [a,b] such that f(c)=0. (this only works on the real line?)
Injective iff surjective
- If A and B are finite sets that have the same cardinality, then is injective iff surjective.
- If V and W are finite-dimensional vector spaces with the same dimension, then a linear map is injective iff surjective.
