Strong operator topology

In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form $T\mapsto\|Tx\|$ , as x varies in H.

Equivalently, it is the coarsest topology such that the evaluation maps $T\mapsto Tx$ (taking values in H) are continuous for each fixed x in H. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets $U(T_0,x,\epsilon)=\{T:\|Tx-T_0x\|<\epsilon\}$ (where T₀ is any bounded operator on H, x is any vector and ε is any positive real number).

单词	Strong operator topology
释义	Strong operator topology 英语百科 Strong operator topology In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form $T\mapsto\\|Tx\\|$ , as x varies in H. Equivalently, it is the coarsest topology such that the evaluation maps $T\mapsto Tx$ (taking values in H) are continuous for each fixed x in H. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets $U(T_0,x,\epsilon)=\{T:\\|Tx-T_0x\\|<\epsilon\}$ (where T₀ is any bounded operator on H, x is any vector and ε is any positive real number).
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