Operators which are not bounded will not be defined in the whole Hilbert space ℌ, but only in a subspace of b , called the “domain” of the operator and denoted by b A. Operators defined only in a subspace of b occur quite frequently. Of course, integral operators are naturally defined only in such subspaces of b. But if they are bounded — and those that we have considered are bounded — they can be extended to the whole space ℌ. Differential operators also are defined only in subspaces — as was already indicated in Chapter I; but they are always strictly non-bounded, as will be shown in Chapter VII. Of course one will naturally try to extend the domain of a non-bounded operator as far as possible.
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