Algorithmic Stability in Infinite Dimensions: Characterizing Unconditional Convergence in Banach Spaces

The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers theorem establishes their strict separation in general Banach spaces. We present a comprehensive characterization theorem unifying seven equivalent conditions for unconditional convergence: permutation invariance, net convergence, subseries tests, sign stability, bounded multiplier properties, and weak uniform convergence. These theoretical results directly inform algorithmic stability analysis, governing permutation invariance in gradient accumulation for Stochastic Gradient Descent and justifying coefficient thresholding in frame-based signal processing. Our work bridges classical functional analysis with contemporary computational practice, providing rigorous foundations for order-independent and numerically robust summation processes.