Vague Vector Spaces, Vague Normed Spaces, Vague Banach Spaces and Vague Hilbert Spaces: A Unified Mathematical Framework

Classical vector spaces, normed spaces, Banach spaces and Hilbert spaces assume that algebraic membership and metric information are precisely specified. In applications involving incomplete measurements, uncertain coefficients, noisy data and approximate computational states, such precision may not be available. This paper develops a coherent interval-valued vague framework for these structures. A vague value is represented by a truth degree and a falsity degree, while a vague magnitude is represented by a closed interval of nonnegative real numbers. Vague vector spaces are introduced by attaching compatible truth and falsity functions to an ordinary vector space. Vague norms are then defined through lower and upper magnitude functions satisfying positivity, definiteness, absolute homogeneity and componentwise triangle inequalities. Vague convergence, vague Cauchy sequences and vague completeness are formulated using the upper norm. A vague Banach space is defined as a complete vague normed vector space. A corresponding vague inner-product framework is developed, leading to the definition of vague Hilbert spaces. A principal construction is obtained from any classical normed space by the uncertainty envelope ||x||_v=[(1-epsilon)||x||,(1+epsilon)||x||], 0<=epsilon<1. It is proved that this construction preserves convergence, Cauchy sequences and completeness. Finite-dimensional spaces, sequence spaces, continuous-function spaces, L2 spaces and matrix spaces are used as examples. The relationship with classical Banach and Hilbert spaces is established, and applications to uncertain approximation, digital mathematics, matrix analysis and quantum-inspired computation are discussed. The paper also identifies limitations of interval-valued inner products and directions for further research.