Direct Summation of the same space , meaning? In a course , there is the definition of the notation :(for V a G module over a finite group)
$$V^{\oplus n}=V\oplus V\oplus ...\oplus V\quad(\text{n times})$$
But i don't understand the sense of this object , for $x\in V^{\oplus n}$ there is no unicity for the writing x=$v_1+v_2+...+v_n$ so what's the meaning of the direct sum , i don't get it,
 A: This is called the exterior direct sum. As a vector space, it is the Cartesian product $V^n$ equipped with componentwise addition. Though there is a slight difference where infinite direct sums are concerned. In general,
$$\bigoplus_{i\in I} V_i=\{(v_i)_{i\in I}~\vert~v_i=0\textrm{ for all but finitely many }i\in I\}$$
equipped with componentwise addition and $\lambda(v_i)_{i\in I}=(\lambda v_i)_{i\in I}$ as multiplication.
But the idea is that the exterior direct sum is a vector space into which you can embed all the summands such that the space is the interior direct sum (the one you're familiar with) of the embedded spaces. In this way, we have, for instance $\mathbb R^2\cong \mathbb R\oplus\mathbb R$, since we can embed the first instance of $\mathbb R$ into $\mathbb R^2$ like this: $r\mapsto(r,0)$, and the second instance like this: $r\mapsto(0,r)$. The corresponding images are subspaces of $\mathbb R^2$, and $\mathbb R^2$ is the (interior) direct sum of those subspaces. And then we say that it is the exterior direct sum of two instances of $\mathbb R$ . Basically, we identify the exterior direct summands with fitting interior direct summands of their exterior direct sum.
