Sentence examples for bounded sequence on a from inspiring English sources
Let ({x_{n}}) be a bounded sequence on a reflexive Banach space X.
We show that ((g_{k})) be a bounded sequence on ([0, n]).
Rather than considering the whole space of sequences on X, we shall be working with (ell _{infty}(X)), the usual space of bounded sequences on X, which becomes a Banach space under the sup norm: (Vert{ mathbf {x}}Vert_{infty}= sup_{n}Vert{ mathbf {x}(n)}Vert_{X}).
where { h ( n ) } is a bounded sequence defined on the set of nonnegative integers Z +.
Since is a bounded sequence, it contains a convergent subsequence.
H ( n ) is a nonnegative sequence defined on Z + such that ∑ n = 0 ∞ H ( n ) = 1 and x ( n ) is a nonnegative bounded sequence defined on Z with x ∗ ≤ lim inf n → ∞ x ( n ) ≤ lim sup n → ∞ x ( n ) ≤ x ∗, where x ∗, x ∗ are nonnegative constants.
(ii) If { u n } is a bounded sequence in E, then a function g on E defined by g ( x ) = lim sup n → ∞ ∥ u n − x ∥ (3) .
If { u n } is a bounded sequence in E, then a function g on E defined by g ( x ) = lim sup n → ∞ ∥ u n − x ∥ (3). is strictly quasiconvex, that is, g ( λ x + ( 1 − λ ) y ) < max { g ( x ), g ( y ) }. for all λ ∈ ( 0, 1 ) and x, y ∈ E with x ≠ y.
(C') If a uniformly bounded sequence in converges to a function compactly on (i.e., converges on any compact discrete interval in ) in the compact-open topology, then belong to and as. .
