The Space of Coefficients in a Linear Topological Space

It is proved that the arbitrary nondegenerate system in a linear complete topological space has a correspondence complete topological space of coefficients with canonical basis. Basicity criterion for systems in such spaces is given in terms of coefficient operator.


Introduction
The concept of the space of coefficients belongs to the theory of bases.As is known, every basis in a Banach space has a Banach space of coefficients which is isomorphic to an initial one (see, e.g., Dremin, Ivanov, & Nechitailo, 2001;Singer, 1970;Singer, 1981).Every nondegenerate system (to be defined later) in a Banach space generates the corresponding Banach space of coefficients with canonical basis (see, e.g., Bilalov & Najafov, 2011;Dremin, Ivanov, & Nechitailo, 2001).Similar results are obtained in fuzzy structures (Bilalov, Farahani, & Guliyeva, 2012) which have recently gained a great scientific interest.Therefore, space of coefficients plays an important role in the study of approximative properties of systems.It has very important applications in various fields of science, such as solid body physics, molecular physics, multiple production of particles, aviation, medicine, biology, data compression, etc (see, e.g., Chui, 1992;Edwards, 1969 and references within).All these applications are closely related to wavelet analysis, and there arose a great interest in them lately (see, e.g., Chui, 1992;Christensen, 2003Christensen, & 2004;;Dremin, Ivanov, & Nechitailo, 2001).It is well known that many topological spaces are nonnormable.Therefore, the study of various properties of the space of coefficients in topological and, in particular, in metric spaces is of special scientific interest.
Our work is dedicated to the study of topological properties of the space of coefficients generated by nondegenerate system in a Hausdorff linear topological spaces.It is structured as follows.In Section 2 we state basic concepts and facts to be used later.In Section 3 we state and prove our main results.We prove that the arbitrary nondegenerate system in a linear complete topological space generates complete linear topological space of coefficients with canonical basis.Basicity criterion for systems in such spaces is given in terms of coefficient operator.

Needful Concepts and Facts
We will use the usual notations: N will be the set of all positive integers, R will denote the set of all real numbers, C will stand for the set of all complex numbers, L (X; Y) will be the linear space of continuous linear operators from X to Y, and F will denote the field of scalars.By the linear topological space (X; τ) (LTS in short) we mean the linear space X over field F (F ≡ R or F ≡ C) with a topology τ, where linear operations are continuous and every point is a closed set.Set M ⊂ X is said to be bounded if for an arbitrary neighborhood of zero O ε , ∃δ > 0 : λM ⊂ O ε , ∀λ : |λ| < δ.By the neighborhood O ε (x 0 ) of point x 0 ∈ X we mean an open set x 0 +O ε , where O ε is a neighborhood of zero.Local base in X is a family of the neighborhoods of zero B such that the every neighborhood of zero contains some neighborhood belonging to B. (X; τ) is called a metrizable space if its topology τ is generated some metric.Metrizable and complete space is called Frechet space or F -space.By L [M] we denote the linear span of the set M ⊂ X.The closure of this set in topology τ is denoted by M. We also state some facts from the theory of LTS which will be used later in this work.
Let M be some set and " ≤ " an order relation (partial Let (X; τ) be some LTS.Net {x λ } λ∈M converges to k 0 ∈ X over M, if for any neighborhood O x 0 of the point x 0 , ∃λ 0 ∈ M : x λ ∈ O x 0 , ∀λ ≥ λ 0 .This fact we will denote by lim If any Cauchy net converges in (X; τ), then this space is called complete.
We will use the following concept.
Also recall some facts which will be needed to obtain our main results.
Theorem 1 (Chui, 1992) If B is a local base in LTS (X; τ), then every neighborhood of zero belonging to it contains the closure of some other neighborhood of zero in B.
is called the covering of the set M, where A is an index set.More details of these and other facts and concepts can be found in the monographs (Bourbaki, 1959(Bourbaki, & 1968;;Edwards, 1969;Heil, 2011;Rudin, 1975).

Space of Coefficients
Let (X; τ) be a complete linear topological space and let {x n } n∈N ⊂ X be some nondegenerate system.Assume Obviously, K x is a linear space with regard to usual operations of component-specific addition and multiplication by a scalar.Every neighborhood of zero O ε in X generates corresponding neighborhood of zero O K ε in K x: The set of neighborhoods of zero O K ε in K x generates corresponding topology τ K in K x.Every linear topological space is a Hausdorff space.Consequently, every compact in such spaces is bounded.Therefore, every Cauchy sequence is pre-compact and, consequently, is bounded in such spaces.More details about these facts can be found in Bourbaki (1959) and Heil (2011).
Let us show that (K x; τ K ) is complete.We will need the following: Proof.Suppose that the net { f λ } λ∈M does not converge to zero.Also suppose that { f λ } λ∈M has a bounded subsequence λ n k k∈N .Then it is possible to derive a convergent subsequence from it, and, without loss of generality, we will assume that λ n k → λ 0 , k → ∞.We have λ n k x → λ 0 x as k → ∞.Hence, λ 0 = 0, because (X; τ) is a Hausdorff space (see, e.g., Bourbaki, 1968).Thus, every bounded subsequence {λ n } n∈N converges to zero.It follows from the assumption made above that {λ n } n∈N contains unbounded subsequence.Let λ n k x = x 0. So we came upon a contradiction which proves the lemma.
Take an arbitrary Cauchy net fλ λ∈M ⊂K x with fλ ≡ f (λ)   k k∈N . Take a neighborhood of zero O K ε in K x generated by an arbitrary neighborhood of zero O ε in X. Definitionally this means that ∃λ 0 ∈ M : fλ And this, in turn, means that the net is convergent in X, and, as a result, f (λ)   2 λ∈M is convergent.
Continuing this process, we obtain that f (λ) Let us show that f ∈K x and fn → f as λ ∈ M. Let B be some local base of the neighborhoods of zero in X.By virtue of Theorem 2, such base always exists in LTS.Take According to the definition, this implies It follows from the arbitrariness of O ε 1 that the net r k=1 f (λ) k x k λ∈M is Cauchy net in X for ∀r ∈ N. Passing to the limit in (1) as μ ∈ M yields In what follows, we will use the following assertion: Assertion 1 (Rudin, 1975) For every neighborhood of zero W in X there exists a symmetric neighborhood of zero U(in the sense that U = −U) which satisfies the relation U + U + U ⊂ W.
By virtue of Theorem 1, we have directly from this assertion the following.
Corollary 1 For every neighborhood of zero W in X there exists a neighborhood of zero Usuch that Ū + Ū + Ū ⊂ W.
Taking neighborhood W as O ε 1 and assuming O ε 2 = U, we have As fλ ∈K x , it is clear that ∃r 0 ∈ N : We have Taking n ≥ m 0 , by virtue of inclusions ( 2)-( 4) we get Let us show that the linear operations are continuous in K x.Take ∀ f ∈K x and assume a → a 0 in F. Let us show that (a − a 0 ) f → 0 as a → a 0 .Let S m = m n=1 f n x n , with f ≡ { f n } n∈N .As the sequence {S m } m∈N is convergent in X, it is evidently bounded there.Take an arbitrary neighborhood of zero O ε in X.Then ∃δ > 0 : Obviously, ∃r ∈ N : f r 0. Set n 0 = min {r : f r 0}.As f n 0 x n 0 0, there exists a neighborhood of zero O ε in X such that f n 0 x n 0 O ε .Let this neighborhood be corresponded by a neighborhood of zero O K ε inK x: It follows directly that the space K x is a Hausdorff space.So we obtain that every one-point set in K x is closed.Thus, we have proved the following: Theorem 3 Space K x with a topology τ K has the following properties: 1) it is complete; 2) every one-point set in it is closed; 3) linear operations are continuous in it.
Let's consider the operator T :K x → X defined by Take an arbitrary neighborhood of zero O ε 1 in X and consider another neighborhood of zero Passing to the limit in this relation as m → ∞ yields It is clear that if the system {x n } n∈N is ω-linearly independent, then f n = 0, ∀n ∈ N, and, as a result, KerT = {0}.In this case there exists an inverse operator T −1 : ImT →K x.
Denote by {e n } n∈N ⊂K x a canonical system with e n = {δ nk } k∈N , where δ nk is the Kronecker symbol.Let us show that {e n } n∈N forms a basis for K x.Take ∀ f ≡ { f n } n∈N ∈K x and prove that the series ∞ n=1 f n e n is convergent in K x.Take an arbitrary neighborhood of zero O K ε in K x, generated by a neighborhood of zero O ε in X.As the series ∞ n=1 f n x n is convergent in X, we have ∃m 0 ∈ N: For partial sums S r corresponding to the element m+p n=m f n e n we have Therefore it is evident that S r ∈ O ε , ∀r ∈ N. As a result, the series ∞ n=1 f n e n is fundamental in K x and, consequently, is convergent in it.Assume fm = f − m n=1 λ n e n = {...; 0; f m+1 ; ...}.Let O K ε be an arbitrary neighborhood of zero K x, generated by the neighborhood of zero O ε in X.The convergence of the series ∞ n=1 f n x n implies that ∃m 0 ∈ N: m+p n=m f n x n ∈ O ε , ∀m ≥ m 0 , ∀p ∈ N.For partial sums S r corresponding to the element fm we have As a result, we get fm Theorem 4 Let (X; τ) be complete LTS and let {x n } n∈N ⊂ X be some nondegenerate system.Then the corresponding space of coefficients K x; τ K x is also a complete LTS with canonical basis.Now suppose that the system {x n } n∈N is ω-linearly independent and the value area of operator T is closed, i.e.ImT = ImT .It is easily seen that T e n = x n , ∀n ∈ N. Then it is clear that the system {x n } n∈N forms a basis for ImT .Indeed, it directly follows from the definition of the basis and from the expresiion of the operator T .If this system is complete in X, then it forms a basis also for X.We will call T a coefficient operator.Converse is also true, i.e. if the system {x n } n∈N forms a basis for X, then it is clear that it is complete in it and w−linearly independent.It is evident that in this case ImT = ImT .Hence, it is easy to see that ImT = X.Thus, the following theorem is true.
Theorem 5 Let (X; τ) be complete LTS, {x n } n∈N ⊂ X be a nondegenerate system, K x; τ K x be the corresponding space of coefficients, and T :K x → X be the corresponding coefficient operator.System {x n } n∈N forms a basis for X if and only if the following conditions are satisfied: 1) it is complete in X; 2) it is ω-linearly independent; 3) ImT = ImT .

Conclusion
Summing up, we arrive at the following conclusions: 1) In a complete topological vector space every non-degenerate system generates a similar space of coefficients; 2) The space of coefficients has a canonical basis, regardless of whether the system is complete, minimal or forms a basis; 3) A basicity criterion, different from the one for classical case, is obtained in terms of coefficient operator.
and therefore, e * k is continuous ∀k ∈ N. It is easy to see that e * n (e k ) = δ nk , ∀n, k ∈ N. As a result, we obtain that e * n n∈N is a system biorthogonal to {e n } n∈N .This proves the basicity of system {e n } n∈N in K x .So the following theorem is true.