Let $f_n : \mathbb C^n \to \mathbb R^{2n}$ be defined by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$
I am having trouble believing that
$$ \langle X, Y\rangle_{\mathbb C} = \langle f(X) , f(Y)\rangle_{\mathbb R}+ i \langle f(X),f(iY) \rangle_{\mathbb R}$$
The map $f$ is a linear isomorphism. The equation above looks like the right hand side has twice the magnitude (absolute value) of the left hand side. Could someone help me see that they are equal?
(Here the complex inner product is defined as $\langle x,y\rangle = \sum_n x_n \overline{y_n}$)