Smooth norms on certain $C(K)$ spaces

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysi...

Smooth Functions and Partitions of Unity on Certain Banach Spaces

In an earlier paper [4], the author sketched a method, based on the use of “Talagrand operators”, for defining infinitely differentiable equivalent norms on the spaces C0(L) for certain locally compact, scattered spaces L. A special case of this result was that a C renorming exists on C0(L) for every countable locally compact L. Recently, Hájek [3] extended this result by showing that a real no...

NORMS OF CERTAIN OPERATORS ON WEIGHTED `p SPACES AND LORENTZ SEQUENCE SPACES

The problem addressed is the exact determination of the norms of the classical Hilbert, Copson and averaging operators on weighted `p spaces and the corresponding Lorentz sequence spaces d(w, p), with the power weighting sequence wn = n−α or the variant defined by w1 + · · ·+wn = n1−α. Exact values are found in each case except for the averaging operator with wn = n−α, for which estimates deriv...

Smooth biproximity spaces and P-smooth quasi-proximity spaces

The notion of smooth biproximity space  where $delta_1,delta_2$ are gradation proximities defined by Ghanim et al. [10]. In this paper, we show every smooth biproximity space $(X,delta_1,delta_2)$ induces a supra smooth proximity space $delta_{12}$ finer than $delta_1$ and $delta_2$. We study the relationship between $(X,delta_{12})$ and the $FP^*$-separation axioms which had been introduced by...

Norms on Quotient Spaces