The set that you have defined is the (unital) linear subspace generated by x. It is finite-dimensional (in fact, two-dimensional, with a basis consisting of x and e) and therefore it is complete. But it need not be closed under multiplication. The Banach subalgebra generated by x must contain all the powers of x. Therefore it must also contain the set of all polynomials (for all ). The set of all these polynomials is a subalgebra, but it need not be complete. So you then need to take its closure. That way, you get the Banach subalgebra generated by x.