algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
For a linear operator between Banach spaces, compactness is a natural strengthening of continuity (boundedness).
A linear operator is compact if it sends bounded subsets to relatively compact subsets.
For linear operators on Hilbert spaces, instead of compact operators one sometimes speaks of completely continuous operators. However, in the full generality of Banach spaces, by a completely continuous operator one means slightly less: an operator that maps every weakly convergent? sequence to a norm convergent? sequence. For Hilbert spaces and, more generally, for reflexive Banach spaces, the two notions are equivalent.
Since every relatively compact subspace (in a Banach space, or indeed in any metric space) is bounded, every compact operator is bounded. Instead of checking compactness on all bounded subsets it is sufficient to check it for a ball of one fixed radius: an operator is compact iff it sends the ball of unit radius to a relatively compact set.
The quotient of the bounded operators by the compact operators is called the Calkin algebra.
Last revised on June 17, 2025 at 19:27:38. See the history of this page for a list of all contributions to it.