I'm learning optimization and I came across equality constraints.
Lets say we have an objective function J defined over $V$ with values in $\mathbb{R}$, where $V$ is the normed vector space in which the problem lies.
Now we have $v \in K$ where $K \subset V$, and the equality constraint $F(v) = $0, where $F : V \rightarrow \mathbb{R}^m$ ($m$ real constraints: $F_i(v) = 0 )$
Two things I don't understand:
Why does the constraint map the normed vector space to the real number space to the power m, why not to the power one ?
The index $i$ means that we may have different contraints at different directions in space?