(I'll explain what I know first and then I'll ask the questions). Given a finite dimensional vector space $V$, it is often remarked that there is no "natural" isomorphism from $V$ to $V^*$ (I guess this means a basis independent isomorphism?). I understand that one typically constructs an isomorphism $V \to V^*$ by fixing a basis for $V$, call this $B = \{v_1, v_2, \ldots, v_n\}$, and then mapping $v_i \mapsto \delta_i$ where $\delta_i$ is the linear functional given by $\delta_i(v_i) = \delta_{ij}$. Here are my questions:
(1) Why is there no natural isomorphism $V \to V^*$?
(2) If $V$ is a finite dimensional inner product space, we can map each $v \mapsto \langle v, \cdot \rangle$. Is this not "natural"? (I know we can consider sesquilinear forms, but let's keep the discussion to this inner product for simplicity).