The following is stated in the book Analysis on Manifolds by James Munkres
Just as is the case with linear transformations, a multilinear transformation is entirely determined once one knows its values on basis elements. That we now prove.
And then he gives the following lemma.
Lemma 26.2 Let $a_1, \dots, a_n$ be a basis for $V$. If $f, g : V^k \to \mathbb{R}$ are $k$-tensors on $V$ and if $$f\left(a_{i_1}, \dots, a_{i_k}\right) = g\left(a_{i_1}, \dots, a_{i_k}\right) $$ for every $k$-tuple $I = (i_1, \dots, i_k)$ of integers from the set $\{1, \dots, n\}$ then $f=g$.
Now what I don't understand is why to we even need the following in the above lemma
"for every $k$-tuple $I = (i_1, \dots, i_k)$ of integers from the set $\{1, \dots, n\}$"
Because $V^k$ has dimension $k \cdot n$ since $V$ has dimension $n$, and has as basis elements \begin{align*}&(a_1, 0, \dots, 0), \dots, &(a_n,0, \dots, 0), \\ &(0, a_1, \dots, 0), \dots, &(0, a_n, \dots, 0), \\ \ \ \ &. &. \\ \ \ \ &. &. \\ \ \ \ &. &. \\ &(0, 0, \dots, a_1), \dots, &(0, 0, \dots, a_n) \end{align*}
So if $\mathcal{B}$ was the set of basis elements of $V^k$ above then I'd say that the following proposed lemma would make more sense
Proposed Lemma: Let $V$ be a vector space of dimension $n$. If $f, g : V^k \to \mathbb{R}$ are $k$-tensors on $V$ and if $$f(\alpha) = g(\alpha)$$ for every $\alpha \in \mathcal{B}$ where $\mathcal{B}$ is a basis for $V^k$, then $f= g$
Furthermore the same part of Lemma 26.2
"for every $k$-tuple $I = (i_1, \dots, i_k)$ of integers from the set $\{1, \dots, n\}$"
Taken literally gives $n^k$ possible $k$-tuples, which would correspond to checking to see if the values of $f$ and $g$ agree on $n^k$ basis elements, which confuses me since $\dim(V^k) = kn$
I'm sure that Lemma 26.2 must be correct and I'm just making some error somewhere, if so could someone please point out what that error is.
The dimension of $\underbrace{V\otimes\dots\otimes V}_{k \text{ times}} = \bigotimes^k V$ is $n^k$, even though the dimension of $V^k$ is $kn$. We're talking about multilinear maps on $V^k$, not linear maps. (Note that $n=k=2$ is a bad example to pick, since then $kn = n^k$. :))
EDIT: I should comment that the vector space of multilinear maps on $V^k$ is isomorphic to $\big({}\bigotimes^k V\big)^*$, but dimensions are the same.