A classical Banach space: $l_\infty/c_0$

Abstract

This is an expository paper in which we study some of the structural and geometric properties of the Banach space l"/c 0 using its identification with C(j8N\N). In particular, it is noted that although LJCQ is not a dual space, its unit ball has an abundance of extreme points. Also, its smooth points are classified and its complemented subspaces are studied. Introduction.The Banach space ljc 0 certainly falls into the category of a "classical Banach space". Not only has it been around since the time of Banach's original monograph (1932) [2], but it is also classical in the sense of Lacey [17] or Lindenstrauss and Tzafriri [18] since it is congruent (isometrically isomorphic) to the space C(/3N\N). However, many of its interesting properties have not been as widely circulated as those of some of the other classical Banach spaces. In this paper, which is of an expository nature, it is our intention to begin to rectify this situation.We will begin with some definitions. /^ is the linear space of bounded sequences of real numbers and c 0 is the subspace of sequences which converge to zero. Both of these spaces, when provided with the supremum norm, ||JC|| = sup|jtj where x = {x n } n ^l9 are Banach spaces. The quotient space IJCQ is the usual linear space of cosets x = x + c 0 , x e / oe . When provided with the quotient norm ||x|| = inf{||jc -y\\ : y e c 0 }, x = x + c 0 , it is a Banach space.Although much is known about quotient spaces in general, this is not the appropriate method for studying ljc$. Indeed, much more is known about C(JT), the space of continuous real valued functions on the compact, Hausdorff space T, and we shall see shortly that IJCQ is congruent to C(/3N\N).Recall that if T is a TychonofF space, (i.e., completely regular and Hausdorff) then its Stone-Cech compactification ßT can be described as AMS (1980) subject classification : 46B25,46B20.

PFA and complemented subspaces of ℓ∞/c0

“…This implies (Todorčević [13,Theorem 2] and Shelah and Steprāns [11,Lemma 2]) there is a proper poset Q which introduces an uncountable Γ ⊂ ω 1 so that the family {b γ , e γ : γ ∈ Γ} is a Luzin family (it is unsplit in any proper forcing extension). Now, we meet ω 1 many dense subsets of ω <ω 1 2 * P ω 1 * Q in order to decide on the generic function f = f ω 1 added by P ω 1 , and the Luzin gap {b γ , e γ : γ ∈ Γ} as well as the basic properties of the family as detailed in items (1) - (6). Notice that (by the inclusion of ω 1 many dense subsets of P ω 1 ) f · 1 dγ is almost equal to f γ .…”

“…Our paper is motivated by the question (Drewnowski and Roberts [3], Leonard and Whitfield [6]) of whether or not C(N * ) is primary. A Banach space X is primary if whenever X is written as the sum A ⊕ B of complemented subspaces, then one of A, B is isomorphic to X .…”

The classical Banach spaces ℓ_{𝑣𝑎𝑟𝑝ℎ𝑖}/ℎ_{𝜑}

“…(1) This follows from the fact that C(βN\N) is not realcompact (see [13, p. 146], [3]) and cannot be renormed in order to be rotund or smooth (see [2], [10]). …”

Geometria dos espaços de Banach C([0, &alpha; ], X) para ordinais enumeráveis &alpha;

“…Observação 1.33 Um resultado em [36] nos diz que ∞ (N) e C(βN) são isométricos, ou seja, ∞ (N) = C(βN). O mesmo vale quando substituirmos N por um conjunto infinito enumerável Γ qualquer, ou seja, vale a isometria ∞ (Γ) = C(βΓ).…”