Let $X,Y$ be finite dimensional vector spaces over $\mathbb K$ with bases $E,F$ respectively.
Can any bilinear form $B:X\times Y \to \mathbb K$ be written as $$B(x,y) = \sum_{j=1}^m \alpha _j(x)\beta_j (y), \quad (x,y)\in X\times Y, \tag{1}$$ for some $\alpha _j\in X^*, \beta _j\in Y^*$?
On the one hand, given a map defined by the RHS of (1), it is readily verified the result is linear in both components.
Conversely, we use the bases $E$ and $F$, write $$x = \sum _{i=1}^p x_ie_i\qquad y =\sum_{j=1}^r y_jf_j, $$ it is then suggested that expanding the bilinear form $B$ relative to these bases yields a representation for $B$ of the form (1).
Denote $B(e_i, f_j) = :B_{ij}$. I can see the following: $$B(x,y) = B\left (\sum_{i=1}^p x_ie_i\,;\,\sum_{j=1}^r y_jf_j\right ) = \sum_{i=1}^p\sum_{j=1}^r x_i B_{ij}y_j, $$ but how does one get to (1)? We do have isomorphisms $X\cong X^*$ and $Y\cong Y^*$. Can't immediately see if or how it applies.
Set $\alpha_i(x)=x_i$ and $\beta_i(y)=\sum_jB_{ij}y_j$ These are easily seen to be the required functionals.