Continuous linear operators in Banach spaces

Abstract

Let (X, ∥ ∥ _X ) and (Y, ∥ ∥ _Y ) be normed spaces (real or complex). By B(X → Y) we denote the set of all continuous linear operators which map the space X into the space Y. In the set B(X → Y) we can introduce operations of addition and multiplication by scalars in the following way:

$$\begin{array}{*{20}{c}} {({T_1} + {T_2})(x) = {T_1}(x) + {T_2}(x)} \\ {(aT)(x) = aT(x)} \end{array}$$

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