Bounded operators on Hilbert space are linear transformations that map elements from one Hilbert space to another while preserving the structure of the space. These operators are characterized by a specific property: there exists a constant such that the norm of the operator applied to any element is less than or equal to this constant multiplied by the norm of that element, ensuring that they do not distort the size of vectors in a controlled way. This concept is crucial for understanding functional analysis and quantum mechanics as it relates to type I factors, which often arise in the study of representations of these operators.
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