$\mathscr{L}(\mathcal{H})=$ Set of linear operators from $\mathcal{H}\to \mathcal{H}$. For $T\in \mathscr{L}(\mathcal{H}_A\otimes \mathcal{H}_B)$ specified through $T=\sum\limits_{i,j}\gamma_{i,j}A_i\otimes B_j$, where $A_i\in \mathscr{L}(\mathcal{H}_A)$, $B_i\in \mathscr{L}(\mathcal{H}_B)$, $\gamma_{i,j}\in \mathbb{C}$ we define $$f(T)=\sum_{i,j}\gamma_{i,j}tr(B_j)A_i$$Let $g:\mathscr{L}(\mathcal{H}_A\otimes \mathcal{H}_B) \to \mathscr{L}(\mathcal{H}_A)$ be defined as a map satisfying $$\langle{u}|g(Z)|v\rangle=\sum_k\langle {u\otimes e_k}|Z|{v\otimes e_k}\rangle $$ for any orthonormal basis $\{\langle {e_k}|\mid 1\leq k\leq K\}\subseteq \mathcal{H}_B$.
How can I prove that these two maps are well defined?