We observe that the classical notion of numerical radius gives rise to a smoothness in space bounded linear operators on certain Banach spaces, whenever is norm. characterize Birkhoff-James orthogonality finite-dimensional space, endowed with Some examples are also discussed illustrate geometric differences between norm and usual operator norm, from viewpoint smoothness.

Associated to Birkhoff orthogonality, we study angles in a normed space and present some of their basic properties. We also discuss how decide whether an angle is more acute or obtuse than another. In addition, given two vectors $x$ $y$ space, the formula for `cosine' from which can, principal, compute angle. Some examples will be presented.

We verify in this paper that the linearity and orthogonality structures of a (not necessarily local trivial) Hilbert bundle over a locally compact Hausdorff space Ω determine its unitary structure. In fact, as Hilbert bundles over Ω are exactly Hilbert C0(Ω)-modules, we have a more general set up. A C-linear map θ (not assumed to be bounded) between two Hilbert C∗-modules is said to be “orthogo...

In this paper, we define the line of n-dimensional Euclidean space and we introduce basic properties of affine space on this space. Next, we define the inner product of elements of this space. At the end, we introduce orthogonality of lines of this space. provide the terminology and notation for this paper. (4) (a · b) · x = a · (b · x). (6) a · (x 1 + x 2) = a · x 1 + a · x 2. (7) (a + b) · x ...

We introduce a new stopping-time argument, adapted to handle linear sums of noncompactly-supported functions that satisfy fairly weak decay, smoothness, and cancellation conditions. We use the argument to obtain a new Littlewood-Paley-type result for such sums. 0. Introduction. First, an apology. The title, though correct, is somewhat misleading. It should be “Global almostorthogonality implies...

Kernel methods perform nonlinear learning in high-dimensional reproducing kernel Hilbert spaces (RKHSs). Even though their large model-capacity leads to high representational power, it also incurs substantial risk of overfitting. To alleviate this problem, we propose a new regularization approach, nearorthogonality regularization, which encourages the RKHS functions to be close to being orthogo...

The tomographic description of a quantum state is formulated in an abstract infinite dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then use...

Appendix 2B: Proof that the least-squares minimization of a constrained model is equal to the least-squares minimization of an unconstrained model of projected data on the space where the constraint is valid.. Appendix 1: Orthogonality between the column spaces of the matrices in equation [5] 133 6.8.2 Appendix 2: Proof that the least-squares minimization of a constrained model is equal to the ...