A basis for k-tensors

Question

Let $V$ be a vector space of dimension $n$ with basis $e_1, \dots, e_n$. Let $a_1, \dots, a_n$ be the dual basis for $V^\ast$. Show that a basis for the space $L_k(V)$ of $k$-linear functions on $V$ is $\{a_{i_1} \otimes \cdots \otimes a_{i_k}\}$ for all multi-indices $(i_1, \dots, i_k)$ (not just the strictly ascending multi-indices as for $A_k(L)$). In particular, this shows that $\dim L_k(V) = n^k$.

Answer 1

I will just gives the ideas to prove the result.

• Show that $\{a_{i_1}\otimes\dots\otimes a_{i_k},i_j\in[n],j\in [k]\}$ is linearly independent (you will need to evaluate at $(e_{i_1},\dots,e_{i_n})$).
• Show that it's a generating family. To see that, note that you know a $k$-linear form $T$ if you know the values $T(e_{i_1},\dots,e_{i_k})$ for $i_j\in [n]$, $j\in [k]$.