Sentence examples for bounded sequence in the from inspiring English sources

Indeed, let (x n ) be a bounded sequence in the metric tree M. Define the real-valued function φ U ( x ) = lim n, U d ( x n, x ).

In Remark 1.2, we have said that it is natural to assume that ({x_{n}}) is a bounded sequence in the following result.

(i)A mapping is said to be semiclosed (demiclosed) at zero, if for each bounded sequence in the conditions converges weakly to and converges strongly to imply.

Since X = L 1 ( [ 0, T ] × R ) is a separable Banach space and { v ε n x } is a bounded sequence in the dual space X ∗ = L ∞ ( [ 0, T ] × R ) of X, there exists a subsequence of { v ε n x }, still denoted by { v ε n x }, weakly star convergent to a function u in L ∞ ( [ 0, T ] × R ).

Since X = L 1 ( [ 0, T ] × R ) is a separable Banach space and { u ε n x } is a bounded sequence in the dual space X ∗ = L ∞ ( [ 0, T ] × R ) of X, there exists a subsequence of { u ε n x }, still denoted by { u ε n x }, weakly star convergent to a function v in L ∞ ( [ 0, T ] × R ).

Moreover, since X = L1 [0,T] × R) is a separable Banach space and u ε n x, ρ ε n x are bounded sequences in the dual space X* = L∞([0,T] × R) of X, there are two subsequences of u ε n x, ρ ε n x (still denoted by u ε n x, ρ ε n x ) weak star convergent to two functions U, P ∈ L∞([0,T] × R), respectively.

Let and be two sequences in satisfying the following condition:,, and let be a bounded sequence in satisfying the following conditions: (i), (ii), (iii), (iv).

Let be a bounded sequence in Then the following are equivalent.

If is uniformly convex and is a bounded sequence in then the following statements are equivalent: (a).

(ii) If is a closed convex subset of and if is a bounded sequence in then the asymptotic center of is in (see [17, Proposition 2.1]).

If E is a closed convex subset of a complete (operatorname{CAT}(0)) space and if ({x_{n}}) is a bounded sequence in E, then the asymptotic center of ({x_{n}}) is in E. The concept of quasi-linearization was introduced by Berg and Nikolaev [4].