Here is the question I am trying to solve:
Prove that $\lambda$ is injective.
Here is the definition of the linear map $\lambda$:
Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their tensor product $f \otimes g: U \otimes V \to U' \otimes V'$ by
$$(f \otimes g)( u \otimes v) = f(u) \otimes g(v)$$ for all $u \in U$ and $v \in V.$ This gives rise to a linear map
$\lambda: \operatorname{Hom}(U, U') \otimes \operatorname{Hom}(V, V') \to \operatorname{Hom}(V \otimes U, U' \otimes V')$ defined by $$(\lambda (f \otimes g))(v \otimes u) = f(u) \otimes g(v).$$
Here is an answer given to me here What happens if $(f_1\otimes g_1)(u_1 \otimes v_1) = (f_2 \otimes g_2)(u_2 \otimes v_2)$? :
But I do not understand:
1- why we need an orthonormal basis in the solution? where exactly are we using this idea?
2- What exactly the definition of a Mittag Leffler Module?
Could anyone clarify this to me please?
We DON'T "need" an orthonormal basis but it simplifies the calculation. If $v_1$,$v_2$, $v_3$, ...,$v_n$ is an orthonormal basis and v is any vector then $v= a_1v_1+ a_2v_2+ a_3v_3+ ...+ a_nv_n$ where $a_i$ is the dot product of $v_i$ and $v$.