Net Convergence Equivalent to Absolute Convergence
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Theorem
Let $V$ be a Banach space.
Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.
Then the following two statements are equivalent:
- $(1): \quad \ds \sum_{n \mathop = 1}^\infty \norm {v_n}$ converges (absolute convergence)
- $(2): \quad \ds \sum \set {v_n: n \in \N}$ converges (generalised or net convergence)
Proof
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Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {I}.4$ Exercise $10 - 12$