This gives us uniform convergence of continuous functions ( in t ), so that what it converges to is continuous ( in t ).
42.
Stone starts with an arbitrary compact Hausdorff space and considers the algebra of real-valued continuous functions on, with the topology of uniform convergence.
43.
It is clear from these definitions that uniform convergence of f _ n to f on E implies pointwise convergence for every x \ in E.
44.
*PM : uniform convergence on union interval, id = 9834 new !-- WP guess : uniform convergence on union interval-- Status:
45.
*PM : uniform convergence on union interval, id = 9834 new !-- WP guess : uniform convergence on union interval-- Status:
46.
The theorem shows that the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of X '.
47.
The uniform convergence of more general empirical measures becomes an important property of the "'Glivenko Cantelli classes "'of functions or sets.
48.
However, note that in the authors gave an example of stochastic convex optimization for General Setting of Learning where learnability is not equivalent with uniform convergence.
49.
Thus \ Gamma is a uniform convergence group if and only if its action on \ Theta ( M ) is both properly discontinuous and co-compact.
50.
Since \ mathbb { T } is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology.
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