A sequence ({x_{n}}_{n}) in X is called a Cauchy sequence if, for every (epsilon>0) and (lambda L^-{0_{L^), there exists a positive integer N such that (mathcal{M}_{M,N}(x_{n},x_{m},epsilon)>_{L} mathcal{N}(lambda) ) whenever (n, mgeq N).
A sequence ({x_{n} } subset S) is a Cauchy sequence if for every (varepsilon> 0) and (deltain 0, 1]), there exists (N inmathbb{N}) such that (F(x_{n}, x_{m})(varepsilon) > 1 - lambda), whenever (n,m geq N).
In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from Lp(N,τ) (1⩽p<∞, p≠2) to a Banach space X is automatically continuous, whenever N is a separably acting von Neumann algebra.
As a matter of fact, whenever n = 1, from (3.7), we have.
We may assume a n = 0 whenever n > N. In this case, on setting.
So we obtain the embedding W 1, p ( x ) ↪ C 0 is compact whenever N < p-.
Thus, by Lemma 1, d s ( x n, x m ) ≤ ε whenever n, m > n δ.
To reduce the computation cost, node i does not have to compute L is whenever N is is updated.
