Banach Space - Linear Operators

Linear Operators

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B(X, Y). In infinite-dimensional spaces, not all linear maps are automatically continuous. In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball. Thus, the vector space B(X, Y) can be given the operator norm

With respect to this norm B(X, Y) is a Banach space. This is also true under the less restrictive condition that X be a normed space.

Is X a Banach space, the space B(X) = B(X, X) forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.