I know that in $\mathbb{R}^n$ (and in general $F^n$) you can define an Inner Product in the following way:
Let $X,Y \in \mathbb{R}^n$. Let $C \in \mathbb{R}^n$, and let all the components of $C$ be positive.
Then: $\langle 'X,Y \rangle = \sum_{i=1}^n c_ix_iy_i$ is an Inner Product.
I was wondering if this is the only type of Inner Product in $\mathbb{R}^n$? Can you define an inner product in $\mathbb{R}^n$ that doesn't take this form? Thanks.
There are inner products on $\mathbb{R}^n$ that do not take the form you suggestion. In the inner product $$ \langle x, y \rangle = \sum_i c_i x_i y_i$$ each component in $x$ is matched with exactly one component in $y$. But why can't $x_i$ be matched with some $y_j$ for $i \neq j$? Indeed, one can actually arrange for this. In general
$$ \langle x, y \rangle = x^T A y, $$ where $A$ is a symmetric positive definite matrix, classifies all inner products on $\mathbb{R}^n$. Your example arises as a special case where $A$ is a diagonal matrix, $$ A = \begin{bmatrix} c_1 & & \\ & \ddots & \\ & & c_n \end{bmatrix}. $$