Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., x^c_n$. How to show that with these two operation $V$ is a subspace?
In my opinion, $x+'(-1\cdot 'x)=x+'(x^{-1}_1, ..., x^{-1}_n)=(1, 1, ...)\ne 0$. So how is it possible for it to be a vector space?
In addition $x+'0=0\ne x$.
Could anyone point out my errors?
In this case the identity element of the operation $+'$ is not $0$, but $(1,1,..., 1)$.
The identity element is defined depending on your definition of the operation. If the operation is the natural addition, it is $0$, because $a+0=a$ for all element $a$.
In your case, your operation $+'$ is basically defined as multiplication. The identity element $e$ should be such that $x+'e=x$ for all $x$. You can check easily that it should be $(1,\dots,1)$.
The inverse $x^{-1}$ of an operation $*$ is such that $x*x^{-1}=e$ where $e$ is the identity defined as above. So for addition $+$, the inverse is $-x$ as we understand, since $x+(-x)=0$, where $0$ is the additive identity.