Space of all k-linear maps

Definition

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

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

Here $(V,F)$ is a vector space

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

Proof of claim

[Expand]

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

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TODO: Trivial - can't be bothered right now

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See also

• Multilinear map
• Bilinear map
• Linear map