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The composition of subsystems in quantum theory is defined in terms of a mathematical operation known as the tensor product. We proceed to explain this concept, and to show how it fits in the Hilbert space calculus.
The introduction of the Hilbert space as the essential mathematical structure for the formulation of quantum theory was motivated by the following facts.
Given the description of a quantum state in terms of Hilbert space vectors, physical magnitudes (Heisenberg’s matrices) correspond to linear operators on the Hilbert space. A linear operator (or simply, operator) is a linear map of a Hilbert space to itself.