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.

翻译：无限维空间中条件收敛、无条件收敛与绝对收敛之间的区别对计算算法具有根本性意义。这些概念在有限维空间中相互重合，而Dvoretzky-Rogers定理确立了它们在一般巴拿赫空间中的严格分离。我们提出了一个统一无条件收敛七个等价条件的完备特征定理：置换不变性、网收敛、子级数检验、符号稳定性、有界乘子性质以及弱一致收敛。这些理论结果直接指导算法稳定性分析，既制约着随机梯度下降法中梯度累积的置换不变性，又为基于框架的信号处理中的系数阈值化提供了理论依据。我们的研究架起了经典泛函分析与当代计算实践之间的桥梁，为顺序无关且数值稳健的求和过程奠定了严格的理论基础。