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In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology .
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain. [2]
Uniform convergence implies both local uniform convergence and compact convergence, since both are local notions while uniform convergence is global. If X is locally compact (even in the weakest sense: every point has compact neighborhood), then local uniform convergence is equivalent to compact (uniform) convergence.
Local uniform convergence (i.e. uniform convergence on a neighborhood of each point) Compact (uniform) convergence (i.e. uniform convergence on all compact subsets) further instances of this pattern below. Implications: - "Global" modes of convergence imply the corresponding "local" and "compact" modes of convergence. E.g.:
Hence the sequence f n is uniformly bounded on compact sets. If two subsequences converge to holomorphic limits f and g, then f(0) = g(0) and with f'(0), g'(0) ≥ 0. By the first part and the assumptions it follows that f(D) = g(D). Uniqueness in the Riemann mapping theorem forces f = g, so the original sequence f n is uniformly convergent on ...
By letting be the set of all convex balanced weakly compact subsets of , ′ will have the Mackey topology on ′ or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by (′,) and ′ with this topology is denoted by (′,) ′.
The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on B(H). It is stronger than all the other topologies below. The weak (Banach space) topology is σ(B(H), B(H) *), in other words the weakest topology such that all elements of the dual B(H) * are continuous.