How can I make sense on an inner product of two vectors expressed in different bases? For example, let $\mathbf {(a,b,c)}$ and $\mathbf {(a^*,b^*,c^*)}$ be two basis for the same three dimensional vector space $V$.
Now, let $$\mathbf v=v_1 \mathbf a+v_2\mathbf b+v_3\mathbf c,$$ and $$\mathbf u=u_1 \mathbf a^*+u_2\mathbf b^*+u_3\mathbf c^*$$ be two vectors belonging to V.
What is the meaning of taking the following inner product:
$$\mathbf v \cdot \mathbf u = ?$$
Update:
Consider the case $\mathbf v, \mathbf u \in \mathbb Z^3$, how can I choose the second basis $\mathbf {(a^*,b^*,c^*)}$ such that, the product
$$\mathbf v \cdot \mathbf u = 2\pi n, \;\;\;\;\; n \in \mathbb Z$$
in some sense?
The sum $$ aa^* + bb^* + cc^* $$ has no meaning since its value depends on the bases.
If you know how to express one of the bases in terms of the other you can compute the inner product of those two vectors. You will probably want the two bases to be orthonormal.