Space of all k-linear maps

Definition

For a [ilmath]k\in\mathbb{N} [/ilmath] and a family [ilmath]U_1,\cdots,U_k[/ilmath], of vector spaces over a field [ilmath]F[/ilmath] we denote[1]:

• [ilmath]L(U_1,\cdots,U_k)[/ilmath] as the space of all [ilmath]k[/ilmath]-linear maps with domain [ilmath]U_1\times\cdots\times U_k[/ilmath] that map to any vector space (over [ilmath]F[/ilmath])

Here [ilmath](V,F)[/ilmath] is a vector space

• [ilmath]L(U_1,\cdots,U_k;V)[/ilmath] is the space of all [ilmath]k[/ilmath]-linear maps of the form [ilmath]:U_1\times\cdots\times U_k\rightarrow V[/ilmath]
• Claim: [ilmath]L(U_1,\cdots,U_k;V)[/ilmath] is a vector space. For [ilmath]f,g\in L(U_1,\cdots,U_k;V)[/ilmath] and [ilmath]\lambda\in F[/ilmath] we define the operations as:
• [ilmath](f+g)(x_1,\cdots,x_k)=f(x_1,\cdots,x_k)+g(x_1,\cdots,x_k)[/ilmath] and
• [ilmath](\lambda f)(x_1,\cdots,x_k)=\lambda f(x_1,\cdots,x_k)[/ilmath]

Proof of claim

Claim 1: [ilmath]L(U_1,\cdots,U_k;V)[/ilmath] is a vector space with the operations [ilmath](f+g)(x_1,\cdots,x_k)=f(x_1,\cdots,x_k)+g(x_1,\cdots,x_k)[/ilmath] and [ilmath](\lambda f)(x_1,\cdots,x_k)=\lambda f(x_1,\cdots,x_k)[/ilmath]

TODO: Trivial - can't be bothered right now