View source for The vector space of all linear maps between two spaces

{{DISPLAYTITLE:The vector space of all linear maps between two spaces - {{M|L(U,V)}}}}{{Stub page|grade=A*|msg=Time to start committing to these}}
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==Definition==
Let {{M|\mathbb{K} }} be a [[field]] and let {{M|(U,\mathbb{K})}} and {{M|(V,\mathbb{K})}} be [[vector spaces]] over {{M|\mathbb{K} }}. We define:
* {{M|L(U,V):\eq\{f:U\rightarrow V\in\mathcal{F}(U,V)\ \vert\ f\ \text{is a } }}[[linear map|{{M|\text{linear map} }}]]{{M|\} }}, here {{M|\mathcal{F}(U,V)}} denotes the [[set of all functions from X to Y|set of all functions from {{M|U}} to {{M|V}}]]<ref group="Note">You may have seen this before as {{M|V^U}} - the set of all maps from {{M|U}} into {{M|V}}</ref> - {{XXX|this notation isn't fixed yet}}
That is to say {{M|L(U,V)}} denotes the '''set of all linear maps from {{M|U}} to {{M|V}}'''.
* '''Claim 1: ''' {{M|L(U,V)}} is a [[vector space]] over {{M|\mathbb{K} }} in its own right
==Proof of claims==
===Claim 1: {{M|L(U,V)}} is a vector space over {{M|\mathbb{K} }}===
# Addition operation:
#* Let {{M|f,g\in L(U,V)}} then we define:
#** {{M|(f+g):U\rightarrow V}} by {{M|(f+g):u\mapsto f(u)+g(u)}}
#* Explicitly, the operation {{M|+:L(U,V)\times L(U,V)\rightarrow L(U,V)}} is {{M|+:(f,g)\mapsto(f+g)}} as defined above.
# Scalar multiplication operation:
#* Let {{M|\alpha\in\mathbb{K} }} and let {{M|f\in L(U,V)}} then define:
#** {{M|(\alpha f):U\rightarrow V}} by {{M|(\alpha f):u\mapsto \alpha f(u)}}
#* Explicitly, the operation {{M|*:\mathbb{K}\times L(U,V)\rightarrow L(U,V)}} is {{M|*:(\alpha,f)\mapsto (\alpha f)}} as defined above.
{{Requires proof|easy=yes|msg=It remains to be shown that with these operations that {{M|L(U,V)}} is actually a vector space, however the remainder of the proof is easy and routine|grade=C}}
==See also==
* [[The set of all continuous linear maps between spaces]] - {{M|\mathcal{L}(U,V)}}
* [[The set of all continuous maps between spaces]] - {{M|C(X,Y)}}

==Notes==
<references group="Note"/>
==References==
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{{Requires references|grade=A*|msg=Find some backup}}
{{Definition|Linear Algebra|Functional Analysis}}
{{Theorem Of|Linear Algebra|Functional Analysis}}