algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In a (Hausdorff) topological vector space one can consider bounded sets: a set $S$ is bounded if it is absorbed by any open neighborhood $U$ of zero (i.e. a dilated multiple $\lambda U = \{\lambda x | x\in U\}$ contains $S$). This specializes to the usual definition of a bounded set in a normed vector space: a set is bounded if it is contained in a ball of some finite radius $r \gt 0$.
A linear operator $A: V_1\to V_2$ between topological vector spaces is bounded if it sends each bounded set in $V_1$ to a bounded set in $V_2$. For normed spaces, this is equivalent to saying that it sends the unit ball to a bounded set. Between finite dimensional normed spaces, every linear operator is bounded. A linear operator between any two normed linear spaces is bounded iff it is continuous.
There is also a rich theory for unbounded operators on Hilbert spaces.
The quotient of the bounded operators by the compact operators is called the Calkin algebra.
Every bounded operator on a Hilbert space has a polar decomposition.
The Gelfand-Naimark theorem says that every C*-algebra is isomorphic to a $C^\ast$-algebra of bounded linear operators on a Hilbert space.
Important classes of bounded operators are the compact operators, trace-class operators and Hilbert-Schmidt operators.
bounded linear operator / unbounded linear operator
Last revised on May 5, 2023 at 06:02:45. See the history of this page for a list of all contributions to it.