Throughout H(U,S) will be used for Γ(U,S) to emphasize the cohomological approach we are taking. First it might be a good idea to recall what a Fréchet space is. A seminorm is an object satisfying all of the norm axioms except for the requirement to have a nonzero kernel. A Fréchet space is a vector space F together with a sequence of seminorms {pn} on F such that (i) if pn(f) = 0 for all n, th...

A monotone seminorm p on a Riesz space L is called a Fatou if p(a„)tp(«) holds for every u e L and sequence {u„} in L satisfying 0 < un\u. A monotone seminorm p on L is called strong Fatou if p(u )tp(u) holds for every u e L and directed system {«„} in L satisfying 0 < «„tu. In this paper we determine those Riesz spaces L which have the property that, for any monotone seminorm p on L, the large...

In the error analysis of the process of interpolation by translates of a single basis function, certain spaces of functions arise naturally. These spaces are de ned with respect to a seminorm which is given in terms of the Fourier transform of the function. We call this an indirect seminorm. In certain well-understood cases, the seminorm can be rewritten trivially in terms of the function, rath...

This paper studies the H1 Sobolev seminorm of quadratic functions. The research is motivated by the least-norm interpolation that is widely used in derivative-free optimization. We express the H1 seminorm of a quadratic function explicitly in terms of the Hessian and the gradient when the underlying domain is a ball. The seminorm gives new insights into least-norm interpolation. It clarifies th...

The result stated in the title is proved as a consequence of an appropriate generalization replacing the square property of a seminorm with a similar weaker property which implies an equivalence to the supnorm of all continuous functions on a compact Hausdorff space also. Theorem. Let p be a seminorm with the square property on a complex (associative) algebra A. Then the following hold for all ...

Let A be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be A–continuous if Q̃p ∈ Ainj(X, X̃p), where X̃p is the completion of the normed space Xp = X/p−1(0) and Q̃p is the canonical map. p is said to be a Groth(A)–seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map Q̃pq : X̃q −→ X̃p belongs to A(X̃q, X̃p). It is well-known that when A is the...

Let us consider two compact connected and locally connected Hausdorff spaces M , N and two continuous functions φ : M → R, ψ : N → R . In this paper we introduce new pseudodistances between pairs (M,φ) and (N,ψ) associated with reparametrization invariant seminorms. We study the pseudodistance associated with the seminorm ‖φ‖ = maxφ − minφ, denoted by δΛ, and we find a sharp lower bound for it....

Abstract Our aim is to characterize the homogeneous fractional Sobolev–Slobodecki? spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) ...

We introduce the T -restricted weighted generalized inverse of a singular matrix A with respect to a positive semidefinite matrix T , which defines a seminorm for the space. The new approach proposed is that since T is positive semidefinite, the minimal seminorm solution is considered for all vectors perpendicular to the kernel of T .

The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert’s projective metric or to a related family of seminorms (Hopf’s oscillation or Hilbert’s seminorm). We generalize these properties to abstract consensus operators over normal cones, which include the unital completely positive maps (Kraus operators) arising in...