Can i approximate the inner product of two vectors by the sum of their coefficients? Is there any correlation of the inner product between two $n$ dimensional vectors $\vec x ,\vec y$ with non negative coefficients and the summation of their coefficients? Put it more clearly can i approximate the inner product $\vec x \cdot\vec y$ by the sum of their coefficients: $\sum_{i=0}^{i=n-1}({x_i+y_i})$? Or with any another operation which is homomorphic to the inner product
 A: If this is the standard n-dimensional $\mathbb{R}^n$ the inner product is
$\sum_{i=1}^{n}{x_i y_i}$? Note that your formula misleadingly works with $n+1$ coefficients.
A: Let
$$
\sigma(x,y)=\sum_{i=1}^n|x_i|+|y_i|
$$
Consider the vectors $x=(1,0)$ and $y(0,1)$: $x\cdot y=0$, yet $\sigma(x,y)=2$.
For the other direction, consider $x=(a,0)$ and $y=(a,0)$: $x\cdot y=a^2$ and $\sigma(x,y)=2a$. As $a\to\infty$, the inner product gets far bigger than the sum of the coefficients.
Thus, there can be no constant so that
$$
|x\cdot y|\le C\sigma(x,y)
$$
or so that
$$
\sigma(x,y)\le C|x\cdot y|
$$
even when the coefficients are non-negative.
A: I would say that the sum of the coefficients is an unboundedly poor approximation (that is, the error can grow without bound).  Consider $\langle 0, x \rangle$.  Your approximation yields $\sum_i x_i$, whereas a real inner product would yield $0$.
The question we must ask is, to what end are you trying to "approximate" the inner product?  You might want to look up the Riesz representation theorem.
