The distinction between geometric "spaces" and algebraic "structures" is sometimes clear, sometimes elusive. Clearly, groups are algebraic, while Euclidean spaces are geometric. Modules over rings are as algebraic as groups. In particular, when the ring appears to be a field, the module appears to be a linear space; is it algebraic or geometric? In particular, when it is finite-dimensional, over real numbers, and endowed with inner product, it becomes Euclidean space; now geometric. The (algebraic?) field of real numbers is the same as the (geometric?) real line. Its algebraic closure, the (algebraic?) field of complex numbers, is the same as the (geometric?) complex plane. It is first of all "a place we do analysis" (rather than algebra or geometry).