All vector spaces are by default over or . We consider three vector spaces , and . Let be a tensor product of and , and a tensor product of and . The aim is first to show that , together with the mapping: is a tensor product of , and . Proof:…
Vector spaces and associative algebras will be implicitely all on the same field or . Let be a vector space. We wish to build a free associative algebra on . For any natural number , we will note (tensor product times). Each is a vector space. We may form: An element of can…
Two cases, that I have been unable to unify: a filter base over a power set with a set, or over . First case: filter bases over a power set filter base over , with a set. Definitions Property It is always the case that . Proof: Let be an eventual…
Let and be filters on a same set . Then is also a filter on . Proof: belongs to both and , hence it belongs to , and . Since is not an element of , it is not an element of . Let be an element of and a superset of . Then and…
A category is by definition concrete if there exists a faithful functor from to the category of sets . (A functor is faithful if it is one-to-one (injective) regarding morphisms.) It happens to be the case that in the category , monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections. We…
For the definition and characterization of the polynomials over a field , see Polynomials over a field. Definition of the degree of a polynomial The degree of a polynomial P, written , is an element of . The symbol is taken to have the usual properties of order and of addition relative to natural numbers….
In this context, we consider a commutative field (simply: field) . The polynomials will be constructed as a certain associative unital algebra over (“-aua”), together with a distinguished element called the formal variable. itself, as a vector space over itself and the usual multiplication in as the third law, is a -aua, Morphisms between two…
Definition of a filter Let be a set. Traditional definition of a filter on : A collection of subsets of is a filter on iff: is non-empty (equivalently in context: ); ; ; . It follows from the definition that there can be no filters on the empty set. If we waive the second rule…
In Chapter 31 (“Compact-Open Topology”), the fact that, on the set of the continuous mappings the second topology (the uniform convergence topology) is finer than the first (the compact-open topology) is said to be “clear” but actually needs some non-trivial proof. Part of this is the lemma that I state and prove below. For any …
In Chapter 31, “The Compact-Open Topology”, it is asserted that “clearly” the third topology on (with and the real line) is finer than the second. The issue is more fully discussed here. In this post I discuss the concept of open tubes, particular subsets of the Cartesian product with a set and a metric space. Definition…