Let $(M,g)$ be a Riemannian manifold, $A$ a $(1,1)$-tensor and $\{e_i\}_{i=1}^n$ a local orthonormal frame. This is my calculation for $\sum_i(e_i.f)g(AY,e_i)$:
$$\sum_i(e_i.f)g(AY,e_i)=\sum_i\sum_k(e_i.f)g((AY)^ke_k,e_i)=\sum_i\sum_k(AY)^k(e_i.f)g(e_k,e_i)=\sum_i\sum_k(AY)^k(e_i.f)\delta_{ik}=\sum_i((AY)^ie_i.f)=AY f.$$ where $f$ is a smooth function. Is my calculation correct? What is the geometrical (no algebraically) reason of this equality?