Principal Component Analysis definition
By projecting a vector x using PCA (on the PCs), you maximize the variance in the reduced space. Initially, the space is not optimal in terms of maximizing the variance.

So:


  PCA projects vector  to a space of  dimensions where the
  difference between the initial vector and the projection has maximum
  energy.


(initially the is no maximum variance but after projection the space is optimal in terms of maximizing the variance in the PC defined space)

PS:

Minimizing J1 (the error) corresponds to maximizing the quadratic form e'Se. After demeaning this is e'X'Xe. Set now w = Xe (the projection of the data X). The variance in the projected space is w'w that is equal to the e'X'Xe. Thus, minimizing the error means maximizing varinace