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