Let $X_i$ be finite-dimensional vectors spaces over the same field $F$; $1\leq i \leq k$. How to describe a basis for $\prod_{i=1}^k X_i?$ For example if we consider $\mathbb{R}^3\times \mathbb{R}^3\times \mathbb{R}^3$, the is the following set
$B=\{(e_1, \theta, \theta), (e_2, \theta, \theta), (e_3, \theta, \theta), (\theta, e_1, \theta), (\theta, e_2, \theta), (\theta, e_3, \theta), (\theta, \theta, e_1), (\theta, \theta, e_2), (\theta, \theta, e_3)\},$
$e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1)$, $\theta=(0,0,0)$; a basis of $\mathbb{R}^3\times \mathbb{R}^3\times \mathbb{R}^3$?
A linear map on a finite-dimensional space is determined by its values on a basis. Is the same holds true for a multilinear map? I believe it is true.
However, if $B$ is a basis for $\mathbb{R}^3\times \mathbb{R}^3\times \mathbb{R}^3$ and if $T$ is a multilinear map from $\mathbb{R}^3\times \mathbb{R}^3\times \mathbb{R}^3$ to some vector space $V$, then due to the multilinearity of $T$, $T$ maps each member of $B$ to the zero vector:
$T(x, \theta, \theta)=T(x, 0.\theta, \theta)= 0T(x, \theta, \theta)= \theta'$.
Where am I doing mistake. I apologize, if I made any trivial mistake.
The confusion is due to the fact that a linear map on a product space and a multilinear map on the product of the spaces are different concepts. Consider the following example.
$$f(x,y) = xy.$$
This is a bilinear map. For any $\lambda$, we have that $f(\lambda x, y) = f(x, \lambda y) = \lambda f(x,y) = \lambda xy$. Similarly we have linearity in each component separately.
However, it is clear that $f$ is not a linear map on $\mathbb{R}^2$.
So for your example, if $B_1, B_2, B_3$ are the basis for three spaces. Then $T$ is defined by its values on $B_1 \times B_2 \times B_3$. So $T$ is defined by its values on all 27 triples of the form $(e_i, e_j, e_k)$ for $i,j,k = 1,2,3$.