Matriks Copson dan Ruang Barisan

Let be a linear brownian motion. Traditionally, this brownian motion is considered as a process which belongs a.s. to the space of continuous functions defined on [01], vanishing at the origin and endowed with the topology of uniform convergence. Let now be the besov space of the functions defined on [01], . They are spaces of sequences and an important fact is therefore that these Besov spaces are isomorphic to suitable Banach spaces of numerical sequences. This property is particularly well adapted to the study of brownian motion considered as a process which belongs a.s.to the spaces In this article we illustrate this in the following situations:Z. Ciesielski fias shown recently [C3] that brownian motion belongs a.s. to the space There we characterize in this first paragraph the set indexes αp, qsuch that brownian motion belongs a.s. to The classical theory of large deviations gives the behaviour, as ,of where Γ is a Borel set of the uniform topology. In the second paragraph we show that if Γ are chose...

We establish necessary conditions of optimality for discrete-time infinite-horizon optimal control in the presence of constraints at infinity. These necessary conditions are in form of weak and strong Pontryagin principles. We use a functional analytic framework and multipliers rules in Banach (sequence) spaces. We establish new properties on Nemytskii operators in sequence spaces. We also provide sufficient conditions of optimality.

We introduce and investigate the notion of $p$-convergence in a Banach space. Among others, a Grothendieck-like result is obtained; namely, a subset of a Banach space is relatively $p$-compact if and only if it is contained in the closed convex hull of a $p$-null sequence. We give a description of the topological dual of the space of all $p$-null sequences which is used to characterize the Banach spaces enjoying the property that every relatively $p$-compact subset is relatively $q$-compact ($1\leq q<p$). As an application, Banach spaces satisfying that every relatively $p$-compact set lies inside the range of a vector measure of bounded variation are characterized.

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

The Dual Complexity Space as the Dual of a Normed Cone

In this paper by using natural density real valued bounded sequence space $l_{\infty}$ is extented and statistical bounded sequence space $l_{\infty}^{st}$ is obtained. Besides the main properties of the space $l_{\infty}^{st}$, it is shown that $l_{\infty}^{st}$ is a Banach space with a norm produced with the help of density. Also, it is shown that there is no matrix extension of the space $l_{\infty}$ that its bounded sequences space covers $l_{\infty}^{st}$. Finally, it is shown that the space $l_{\infty}$ is a non-porous subset of $l_{\infty}^{st}$.

A Weak Grothendieck Compactness Principle for Infinite Dimensional Banach Spaces Kaitlin Bjorkman The goal of this thesis is to give an exposition of the following recent result of Freeman, Lennard, Odell, Turett and Randrianantoanina [3] A Banach space has the Schur property if and only if every weakly compact set is contained in the closed convex hull of a weakly null sequence. This result complements an old result of Grothendieck (now called the Grothendieck Compactness Principle) stating that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We include many of the relevant definitions and preliminary results which are required in the proofs of both of these theorems. Dissertation Director: Dr. Kevin Beanland A Weak Grothendieck Compactness Principle for Infinite Dimensional Banach Spaces

In this study will be characterized a matrix that maps the space of the Orlicz sequence space to the space of the bounded sequence and the convergence sequence. This study is carried out by generalized the matrix mapping on lp space, 1&lt;=p&lt;\infty, to the Orlicz sequence space. This results establishes a necessary and sufficient conditions for a matrix that maps the Orlicz sequence space to the bounded sequence space, convergence to sequence space and the convergence sequence space.

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence so that as for every weakly null sequence in X; X has property if every weak* null sequence in admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces.

Let \(E\) be a Banach space (or a Banach lattice), let \(\tau\) be a vector topology on \(E\) and let \({\bf x}\) be a sequence (or a positive sequence) in \(E\) not converging to zero with respect to \(\tau\). We show how to construct infinite dimensional Banach spaces (or Banach lattices) consisting, up to the origin, of sequences in \(E\) not converging to zero with respect to \(\tau\) and containing a subsequence of \({\bf x}\). Plenty of applications to Banach space theory and to Banach lattice theory are provided.

A nuclear operator T on a Banach space is an operator admitting a representation T = (SIGMA) T(,n) where each operator T(,n) has rank one and (SIGMA) (VERT.BAR)(VERT.BAR)(TAU)(,n)(VBAR)(VBAR) < (INFIN). A. Grothendieck proved that the sequence of eigenvalues, repeated according to their multiplicities, of such a nuclear operator must be square summable and that, in fact, two is the best possible exponent of summability. This work uses operators of diagonal type, i.e., operators which admit a diagonal representation with respect to a biorthogonal system, to construct examples of eigenvalue behavior. In particular we show that all of the known results concerning eigenvalue summability of nuclear operators can be obtained using operators of diagonal type. In Chapter 1, we present the terminology and notation to be used in this work. Chapter 2 gives a summary of results concerning nuclear (trace class) operators on Hilbert space. In Chapter 3, some of the important known results concerning eigenvalue behavior of nuclear operators on general Banach spaces are presented. The convolution operators on L(,p)(0,1), 1 < p < (INFIN), p (NOT=) 2, are diagonal operators since the trigonometric system (e('2(pi)inx)) is a conditional basis of these spaces. In Chapter 4 we use these important operators to exhibit extremal behavior of eigenvalue summability of nuclear operators on the L(,p)-spaces. In Chapter 5, we study nuclear cyclic diagonal operators. If X(,1),... ,X(,n) are sequence spaces, an operator D on (CRPLUS)X(,i) is a cyclic diagonal operator if its restriction to each X(,i) is a diagonal operator into X(,i+1) (if i = n into X(,1)). Certain eigenspaces of these operators have important applications to Banach space theory. Finally, in Chapter 6, we use a recently discovered space of G. Pisier to answer affirmatively a long outstanding question of. Pe l cynski and Saphar which is converse to Grothendieck's result: Given a (nonzero) sequence ((lamda)(,n)) in l(,2), is there a Banach space X and a nuclear operator on X whose eigenvalue sequence is ((lamda)(,n))?

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

Generalized locally Toeplitz (GLT) sequences of matrices originated from the spectral study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate either partial differential equations or fractional differential equations using any discretization by local methods (finite differences, finite elements, finite volumes, isogeometric analysis, etc.). The study of the asymptotic spectral behavior of GLT sequences is important in analyzing the solution of the corresponding partial differential equations and in finding fast and efficient methods for the corresponding large linear systems. Approximating classes of sequences (a.c.s.) and spectral symbols are important notions connected to GLT sequences. Recently, G. Barbarino obtained some results regarding the theoretical aspects of such notions. He obtained the completeness of the space of matrix sequences with respect to pseudometric a.c.s. Also, he identified the space of GLT sequences with the space of measurable functions. In this article, we follow the same research line and obtain various connections between the subalgebras of matrix-sequence spaces and the subalgebras of function spaces. In some cases, these are identifications as Banach spaces and some of them are Banach algebra identifications. We also prove that the convergence notions in the sense of eigenvalue/singular value clustering are equivalent to the convergence with respect to the metrics introduced here. These convergence notions are related to the study of preconditioners in the case of matrix/operator sequences. As an application of our main results, we establish a Korovkin-type result in the setting of GLT sequences.