In an NSNLS [mathematical expression not reproducible] are
Cauchy sequences of soft vectors and {[[??].sub.n]} is a
Cauchy sequence of soft scalars in an NSNLS ([??](K), N, *, *), then [mathematical expression not reproducible] are also
Cauchy sequences in NSNLS ([??](K),N, *, *).
Now, we will show that (g([x.sup.1.sub.k])),(g([x.sup.2.sub.k])),..., (g([x.sup.n.sub.k])) are
Cauchy sequences.
For an integer m > 7/2, the solutions obtained in Proposition 8 form the
Cauchy sequences in the following spaces:
As [L.sub.2](J,U), [L.sub.2](],V) are complete spaces, it is sufficient to prove {[u.sub.n,[lambda]]} and {[x.sub.n,[lambda]]} are
Cauchy sequences in [L.sub.2](J, U), [L.sub.2](J, V), respectively
In Section 2 we study some properties of f--statistical convergent and f--statistical
Cauchy sequences, giving a characterization of Banach spaces in those terms.
On statistically convergent and statistically
Cauchy sequences. Indian Journal of Pure and Applied Mathematics, v.
Also we find out the relation between [lambda]-statistical convergent and [lambda]-statistical
Cauchy sequences in this spaces.
122) that there exist
Cauchy sequences which are not Mackey-Cauchy sequences.
As noted in [14], The elements of [[??].sup.p.sub.m] are technically equivalence classes of
Cauchy sequences of elements of [[??].sub.0].
(The difference should not be exaggerated, for indeed the structuralist may welcome constructions based on relatively secure items to help establish coherence or realizability of the defining conditions.) The constructivist takes real numbers as, say,
Cauchy sequences of rationals (either equivalence classes thereof or certain canonical ones), but in the definition of '
Cauchy sequence', constructive quantifiers are to be understood.
We will show that {[z.sub.n]}, {[z'.sub.n]} in X are
Cauchy sequences. Putting [mathematical expression not reproducible] in the equality (18), we get
This shows that ([x.sup.i.sub.k,1]) and ([x.sup.j.sub.1,l])(k, l [less than or equal to] u) are
Cauchy sequences in (X, q).