Intersection Properties of Balls in Banach Spaces

C. R. Jayanarayanan

doi:10.1155/2013/902694

Abstract

We introduce a weaker notion of central subspace called almost central subspace, and we study Banach spaces that belong to the class (GC), introduced by Veselý (1997). In particular, we prove that if <svg style="vertical-align:-0.0pt;width:11.6625px;" id="M1" height="11.175" version="1.1" viewBox="0 0 11.6625 11.175" width="11.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D44C" d="M667 650l-9 -28q-53 -5 -76 -17t-64 -59q-51 -61 -175 -225q-21 -29 -27 -55l-27 -136q-13 -65 -0.5 -80t83.5 -22l-7 -28h-280l8 28q64 4 81 19t30 83l25 128q6 35 -7 65l-98 231q-17 41 -32.5 52t-68.5 16l8 28h252l-6 -28l-40 -4q-27 -3 -33 -12.5t2 -31.5
q8 -26 43 -107.5t61 -134.5q114 145 174 240q14 24 8 33t-37 13l-34 4l8 28h238z" /></g> </svg> is an almost central subspace of a Banach space <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M2" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231
l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z" /></g> </svg> such that <svg style="vertical-align:-0.0pt;width:11.6625px;" id="M3" height="11.175" version="1.1" viewBox="0 0 11.6625 11.175" width="11.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44C"/></g> </svg> is in the class (GC), then <svg style="vertical-align:-0.0pt;width:11.6625px;" id="M4" height="11.175" version="1.1" viewBox="0 0 11.6625 11.175" width="11.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44C"/></g> </svg> is a central subspace of <svg style="vertical-align:-0.0pt;width:28.174999px;" id="M5" height="13.75" version="1.1" viewBox="0 0 28.174999 13.75" width="28.174999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D44B"/></g> <g transform="matrix(.012,-0,0,-.012,13.525,5.525)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5
q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g><g transform="matrix(.012,-0,0,-.012,20.51,5.525)"><use xlink:href="#x2217"/></g> </svg>. We also prove that if <svg style="vertical-align:-0.0pt;width:11.6625px;" id="M6" height="11.175" version="1.1" viewBox="0 0 11.6625 11.175" width="11.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44C"/></g> </svg> is a semi <svg style="vertical-align:-0.0pt;width:17.475px;" id="M7" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g> </svg>-ideal in a Banach space <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M8" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg> such that <svg style="vertical-align:-0.0pt;width:28.262501px;" id="M9" height="14.8" version="1.1" viewBox="0 0 28.262501 14.8" width="28.262501" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,14.737)"><use xlink:href="#x1D44C"/></g> <g transform="matrix(.012,-0,0,-.012,11.6,6.587)"><path id="x27C2" d="M619 0h-566v50h254v498h58v-498h254v-50z" /></g><g transform="matrix(.012,-0,0,-.012,19.584,6.587)"><use xlink:href="#x27C2"/></g> </svg> is an almost central subspace of <svg style="vertical-align:-0.0pt;width:28.174999px;" id="M10" height="13.75" version="1.1" viewBox="0 0 28.174999 13.75" width="28.174999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D44B"/></g> <g transform="matrix(.012,-0,0,-.012,13.525,5.525)"><use xlink:href="#x2217"/></g><g transform="matrix(.012,-0,0,-.012,20.51,5.525)"><use xlink:href="#x2217"/></g> </svg>, then <svg style="vertical-align:-0.0pt;width:11.6625px;" id="M11" height="11.175" version="1.1" viewBox="0 0 11.6625 11.175" width="11.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44C"/></g> </svg> is an <svg style="vertical-align:-0.0pt;width:17.475px;" id="M12" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g> </svg>-ideal in <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M13" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg>. Certain stability results for quotient spaces, injective tensor product spaces, and polyhedral direct sums of Banach spaces are also derived.

Highlights

In [1], Veselystudied a new class of Banach spaces, namely, the class (GC), which were defined in terms of the existence of weighted Chebyshev centers
We prove that properties of being a central subspace and an almost constrained (AC)-subspace are stable under a recently introduced concept called polyhedral direct sums of Banach spaces
Let Y be a subspace of a Banach space X such that Y is an ideal in span{Y, x} for all x ∈ X

Summary

Introduction

In [1], Veselystudied a new class of Banach spaces, namely, the class (GC), which were defined in terms of the existence of weighted Chebyshev centers (see below for definition and details). It follows from [2, Proposition 2.2(a)] that Y is a central subspace of a Banach space X if and only if, for any finite set {yi}ni=1 ⊂ Y and x ∈ X, there exists a y ∈ Y such that ‖y−yi‖ ≤ ‖x − yi‖ for 1 ≤ i ≤ n. We recall that a subspace Y of a Banach space X is called 1-complemented in X if there exists a projection of norm one on X with range Y.

Results

Conclusion