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.

Key words: Continuous compressed sensing, Locally convex spaces, Topological vector space, Hahn-Banach continuous extension theorem, Daniell-Kolmogorov extension theorem, Dual space, Double dual space, Weak-compact and weak-compact spaces, Optimum subspace, Infinite-dimensional vector spaces