Abstract
Few results have been obtained on the packing spheres constant or exact formula for separable Orlicz function spaces (Yang, 2002, P.895-899, Ye, 1987, P.487-493). In this paper, by using the continuity of ideal space norm, we firstly proved that simple function class is dense in L?? function space. This is a necessary condition of interpolation theorem. Hence, the exact value of packing sphere for a class of sparable Orlicz function spaces (with two kinds of norm) is obtained. Secondly, for the space L??[0, 1] discussed in (Yang, 2002, P.895-899), we propose the following conjecture: the L?? [0, 1] space is actually the Lp[0, 1] space, therefore, the results obtained there is actually the proved results in Lp space.
Highlights
It is well known that only finite number of sphere can be packed in a finite dimensional space if the spheres have the same radius but uncrossed, no matter how small is the radius
For infinite dimensional Banach space X, there exist a constant Λ(X) such that infinite number of disjoint spheres can be packed in a unit sphere B(X) if the radius less than Λ(X)
Only finite number of disjoint sphere can be packed in sphere B(X) if the radius larger than Λ(X)
Summary
From Theorem 1.18, we know Orlicz function space LΦ∗ is separable ⇔ Φ ∈ Δ2(∞). We prove that if Φ ∈ Δ2(∞), the two norms || · ||Φ and || · ||(Φ) are continous in function space LΦ∗. We will prove that if simple function class is dense in LΦ∗ , LΦ∗ is seperable, from Proposition 2.1 we have Φ ∈ Δ2(∞). Simple function class is dense in LΦ∗. For function space LΦ∗ [0, 1] normalized by Orlicz norm, its packing sphere value can be estimated by following bound max{. From Property 1.13 and Property 1.14, we can get the above result. For function space LΦ∗ [0, 1] normalized by Luxemburg norm, its packing sphere value can be bounded by max{
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