Putting and consider the sequence.
Take such that and consider the sequence.
(4 if satisfies the condition, for all and considering the sequence one has (1.9).
Let be chosen arbitrarily and consider the sequence generated by (4.1).
Take x0 ∈ X and consider the sequence given by x n + 1 = T x n, n = 0, 1, 2, ….
Let us take any (y= y_{kl})inmathcal{C}_{f}) and consider the sequence (x=(x_{kl})) with respect to the sequence y by equation (1.3) for all (k,linmathbb{N}).
Take a decreasing sequence of positive numbers ({eta_{j}}_{j=0}^{infty}), (eta_{0}leq L_{t_{0}}), such that (lim_{jtoinfty}eta_{j}=0) and consider the sequence of functions (u(t;eta_{j})).
From (10), it follows that the number sequence { ϱ ( θ, F i ) } i = 1 ∞ is fundamental, so it is convergent; denote its limit by r 0 ; obviously, r 0 ⩾ 0. Take an arbitrary radius r > r 0 and consider the sequence { S r F i } i = 1 ∞ ⊂ clos ( X ).
Example 3.8 Let θ = { k r } be a lacunary sequence with lim inf r q r > 1 and consider the sequence x = { x k } as x k = { k, if k is a square ; 0, if k is not a square, i.e., x = { x k } = ( 1, 0, 0, 4, 0, 0, 0, 0, 9, … ).
To show that the strict inclusion may occur, let f be a modulus and consider the sequence (x =(x_{k})) defined by x_{k}= textstylebegin{cases} 1 &mbox{if }k = n^{2}, 0 &mbox{if }k neq n^{2}, end{cases}displaystyle quad n = 1,2,3, ldots.
