Standard notation for sum of vector elements?

$\sum_{i=1}^n v_i$ is the most widely accepted and understood notation.

If you use the euclidean 1-norm and you want the sum of the absolute values of the entries of $\vec{v}$ (or $v_i\geq 0$ ) you can take $\| \vec{v}\|_1$. Otherwise you can use $sum\{\vec{v} \}$ or $trace( diag\{ \vec{v}\})$. But I guess $sum\{\vec{v}\}$ would be the more "standard" notation.

For the Euclidean space $R^n$, where the inner product is given by the dot product: $$<(x_1, x_2, ..., x_n), (y_1, y_2, ..., y_n)> = x_1 y_1 + x_2 y_2 + ... + x_n y_n$$ See Inner Product - Wolfram MathWorld, http://mathworld.wolfram.com/InnerProduct.html

So we can use $\vec{1}$ to form a inner product with vector $\vec{v}$: $$<\vec{1}, \vec{v}> = v_1 + v_2 + ... + v_n$$ or $$<\vec{v}, \vec{1}> = v_1 + v_2 + ... + v_n$$

Pay attention to the dimension of $\vec{1}$, which be the same as that of $\vec{v}$.