A complete metrizable linear space whose

A complete metrizable linear space whose A complete metrizable linear space whose metric cannot be obtained from a norm.

******************************

1. The linear space S consisting of all complex sequences with the metric d((x_i),(y_i)) = å_{i = 1}^¥ [( |x_i - y_i |)/( 2ⁱ (1 + |x_i -y_i|))] is a complete metric space. Since d(2(1,1, ¼),(0,0, ¼)) ¹ 2d((1,1,¼),(0,0, ¼)), the space (S,d) is not normable.

If (X, ||. ||) is a normed linear space then