Abstract
Linear operators from a Hilbert space H into a Hilbert space K are those mappings H → K which are compatible with the vector space structure on both spaces. Similarly, the bounded or continuous linear operators are those which are compatible with both the vector space and the topological structures on both spaces. The fact that a linear map H → K is continuous if, and only if, it is bounded follows easily from Corollary 2.1.1. (A linear map between topological vector spaces is continuous if, and only if, it is continuous at the origin which in turn is equivalent to the linear map being bounded).
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