Abstract
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.
Abstract (translated)
在无限维空间中，条件收敛、无条件收敛和绝对收敛之间的区别对计算算法具有根本性的影响。尽管这些概念在有限维度下是重合的，但Dvoretzky-Rogers定理却证明了它们在一般的Banach空间中存在严格的分离。我们提出了一项全面的特征定理，该定理统一了无条件收敛的七个等价条件：置换不变性、网收敛性、子级数测试、符号稳定性、有界乘法器性质以及弱一致收敛性。这些理论结果直接指导算法稳定性的分析，在随机梯度下降中的梯度累积过程中的置换不变性和基于框架的信号处理中系数阈值的选择方面提供了依据。 我们的工作将经典的功能分析与当代计算实践联系起来，为顺序无关和数值稳健的求和过程提供严格的理论基础。
URL
https://arxiv.org/abs/2601.08512
