Preserving Inner Products and Vector Isomorphism

Definition

1. Let $V,W$ as Inner Product Space with $F_1$
2. $T\in \mathcal{L}(V,W)$ then, $T$ preserves inner products if
3. $⟨T\alpha |T\beta {⟩}_W=⟨\alpha |\beta {⟩}_V,\forall \alpha ,\beta \in V$

This requires the basis of $T$ to be Orthonormal