We obtain refined estimates of the triangle inequality in a normed space using integrals and the Tapia semi-product. The particular case of an inner product space is discussed in more detail.

The spectrum εn has a Brillouin-zone like structure with Ω the width of the first zone, i.e. for any solution |un(t)〉 of quasienergy εn, and any integer m, exp(imΩt)|un(t)〉 is also a solution, with eigenvalue εn + mΩ. The states |un(t)〉 form a Hilbert space of T-periodic functions. The inner product in this space can be defined starting from the standard bra-ket inner product 〈•|•〉 for time-ind...

It is shown that if a representation of a *-algebra on a vector space V is an irreducible *-representation with respect to some inner product on V then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar has suggested using the condition that a given representation of the algebra of quantum observables is a *-representatio...

Let Ω a bounded regular1 domain of R and let us denote respectively by | · | and (·, ·) the natural norm and inner product of the space H = L2(Ω). Let V →֒ H be a real Hilbert space that is continuously and densely embedded into H. Denote by ‖ · ‖ and < ·, · >V respectively the norm and the inner product of V and by L ∈ L(V, V ) the unique linear continuous operator from V to V , the dual of V ,...

We summarize the basic properties of continuous tensor product systems of Hilbert spaces and their role in non-commutative dynamics. These are lecture notes, not intended for publication. 1. Concrete Product Systems In these notes we assume the reader is familiar with the definition and basic properties of E0-semigroups, and has some familiarity with their role in noncommutative dynamics. Our p...

In this paper, we develop a method for unsupervised clustering of twoway (matrix) data by combining two recent innovations from different fields: the Sparse Subspace Clustering (SSC) algorithm [10], which groups points coming from a union of subspaces into their respective subspaces, and the t-product [18], which was introduced to provide a matrix-like multiplication for third order tensors. Ou...

abstract: in the present paper, we give a necessary and sufficient condition for contact cr-warped product to be contact cr-product in kenmotsu space forms.