Compressible Topological Vector Spaces

by   Mohammadreza Robaei, et al.

The optimum subspace decomposition of the infinite-dimensional compressible random processes in the locally convex Hausdorff space has been propose and its dimension has been measured. We conduct topological analysis of finite- and infinite-dimensional compressible vector spaces. We prove that if there are a sufficient number of separating points in compressible topological vector space, the Banach Limit coincides the minimal linear functional. Then, optimum orthogonal decomposition of the compressible topological vector space can be formulated as a limit for which minimal linear functional occurs. It has been shown that the separable space, also referred to as an optimum subspace, is the subset of the compressible vector space that contains its extreme points. The inseparable subspace is characterized using a novel absorbing null space property through Minkowski functional and Lebesgue measure. We prove that the absorbing null space is a connected subspace of the locally convex space. We purpose reflexive homomorphism that establishes a relation between signal space and double dual space. It has been shown that the Banach limit can be determined by the compact convergence of the Cauchy nets on the left and right hands sides of a reflexive homomorphism. Finally, we propose to measure the optimum subspace dimension using the Frechet distance metric. To apply the Frechet distance to the compressible topological vector space, the Kothe sequence sampled from a continuous function has been proposed. Briefly, the proposed approach measures the sufficient number of separating points of the finite- and infinite-dimensional compressible vector space for the given undersampled operator with respect to Hahn-Banach and Daniell-Kolmogorov theorems. The numerical analysis have been presented for finite- and infinite-dimensional signals.


Linear Codes with Prescribed Hull Dimension and Minimum Distance

The hull of a linear code (i.e., a finite field vector space) 𝒞 is defin...

Optimum Distance Flag Codes from Spreads in Network Coding

We study multishot codes in network coding given by families of flags on...

A note on encoding infinity in ZFA with applications to register automata

Working in Zermelo-Fraenkel Set Theory with Atoms over an ω-categorical ...

A Topological Approach to Inferring the Intrinsic Dimension of Convex Sensing Data

We consider a common measurement paradigm, where an unknown subset of an...

Ridges, Neural Networks, and the Radon Transform

A ridge is a function that is characterized by a one-dimensional profile...

Topological Classification in a Wasserstein Distance Based Vector Space

Classification of large and dense networks based on topology is very dif...

Convex Analysis at Infinity: An Introduction to Astral Space

Not all convex functions on ℝ^n have finite minimizers; some can only be...

Please sign up or login with your details

Forgot password? Click here to reset