The Banach Algebra of Bounded Linear Operators

The papers [21], [8], [23], [25], [24], [5], [7], [6], [19], [4], [1], [2], [18], [10], [22], [13], [3], [20], [16], [15], [9], [12], [11], [14], and [17] provide the terminology and notation for this paper. Let X be a non empty set and let f , g be elements of X . Then g · f is an element of X . One can prove the following propositions: (1) Let X, Y , Z be real linear spaces, f be a linear operator from X into Y , and g be a linear operator from Y into Z. Then g · f is a linear operator from X into Z. (2) Let X, Y , Z be real normed spaces, f be a bounded linear operator from X into Y , and g be a bounded linear operator from Y into Z. Then (i) g · f is a bounded linear operator from X into Z, and (ii) for every vector x of X holds ‖(g ·f)(x)‖ ¬ (BdLinOpsNorm(Y, Z))(g) · (BdLinOpsNorm(X,Y ))(f) · ‖x‖ and (BdLinOpsNorm(X, Z))(g · f) ¬ (BdLinOpsNorm(Y, Z))(g) · (BdLinOpsNorm(X, Y ))(f). Let X be a real normed space and let f , g be bounded linear operators from X into X. Then g · f is a bounded linear operator from X into X. Let X be a real normed space and let f , g be elements of BdLinOps(X,X). The functor f+g yields an element of BdLinOps(X,X) and is defined as follows: