We show that B(X), the algebra of bounded linear mappings from normed space X into X is semi-simple:
Suppose that x0 ¹ 0 is fixed in X. Then Ix0 = { T Î B(X) ; Tx0 = 0} is a left ideal in B(X).
We shall show that it is maximal. Let J be a left ideal properly containing Ix0. Then Jx0 = { Tx0 ; T Î J } is a nonzero linear subspace of X which is invariant under each S Î B(X). If
Jx0 ¹ X, then there exists a nonzero y Î Jx0 and an element z Î X such that z Ï Jx0. If S Î B(X) such that Sy = z, then z Î Jx0 for Jx0 is invariant under all elements of B(X). Thus Jx0 = X. So that there exists U Î J such that Ux0 = x0. For each
T Î B(X), TU - UT Î Ix0. Hence T Î J + Ix0 Í J. Therefore B(X) = J.
Thus Rad(B(X)) Í Ç0 ¹ x Î XIx = {0}. Therefore B(X) is algebrically semisimple.