Sentence examples for be a weakly null from inspiring English sources

Let be a weakly null equilateral sequence in and let.

Let ( x n ) be a weakly null sequence in S ( V ϱ ( λ ; p, q ) ).

The binomial sequence space (b_{p}^{r,s}) is of the Banach-Saks type p. Let ((u_{n})) be a weakly null sequence in the (B(b_{p}^{r,s})) unit ball of (b_{p}^{r,s}).

Proof Let ε > 0 and x ∈ Z σ ( s, p ) be such that ∥ x ∥ ≥ ε and ( x n ) be a weakly null sequence in S ( Z σ ( s, p ) ).

Then the space ℓ p ( F ˆ ) has the Banach-Saks type p. Proof Let ( ε n ) be a sequence of positive numbers for which ∑ ε n ≤ 1 / 2, and also let ( x n ) be a weakly null sequence in B ( ℓ p ( F ˆ ) ). Set z 0 = x 0 = 0 and z 1 = x n 1 = x 1.

Assume that is a weakly null sequence in and.

Since is a weakly null net, there exists such that, and for every.

Since the sequence ({ x_{n} } ) is a weakly null sequence, there exists (n_{2}>n_{1}) such that BigglVert sum_{i=1}^{i_{1}}x_{n}(i e_{i} BiggrVert < delta_{0}leq delta_{2}quad mbox{whenever }ngeq n_{2}.

Assume that ((x_{n})) is a weakly null sequence of elements of (B(l_{Phi,p})) with (D((x_{n}))le1).

A Banach lattice is weakly orthogonal if limn→∞∥ |x n |∧|x| ∥ = 0 for all x ∈ X, whenever { x n } n = 1 ∞ Open image in new window is a weakly null sequence, where |x| ∧ |y| = min(|x|, |y|).

If ( x n ) is a weakly null sequence in B X and a ≥ 1, then sup ∥ x ∥ ≤ a [ lim inf n ∥ x + x n ∥ ] = sup ∥ x ∥ = a [ lim inf n ∥ x + x n ∥ ].