Solution

Assume (iii), and let

δ > 0

. Choose

I

such that

\sum_{i \geq I} ∣ ⟨ x_{n}, e_{i} ⟩ ∣^{2} < δ

for every

n

. Coordinate convergence implies, for every finite

J \geq I

\sum_{i = I}^{J} ∣ ⟨ x_{\infty}, e_{i} ⟩ ∣^{2} = lim_{n \to \infty} \sum_{i = I}^{J} ∣ ⟨ x_{n}, e_{i} ⟩ ∣^{2} \leq δ .

Letting

J \to \infty

shows that the corresponding tail of

x_{\infty}

is also at most

δ

∥ x_{n} - x_{\infty} ∥^{2} = i < I \sum ∣ ⟨ x_{n} - x_{\infty}, e_{i} ⟩ ∣^{2} + i \geq I \sum ∣ ⟨ x_{n} - x_{\infty}, e_{i} ⟩ ∣^{2} .

The finite first sum tends to zero by weak coordinate convergence, while the second is at most

2 \sum_{i \geq I} ∣ ⟨ x_{n}, e_{i} ⟩ ∣^{2} + 2 \sum_{i \geq I} ∣ ⟨ x_{\infty}, e_{i} ⟩ ∣^{2} \leq 4 δ .

Since

δ

is arbitrary,

∥ x_{n} - x_{\infty} ∥ \to 0

. Thus (iii) implies (i), completing the equivalence and proving the uniform basis-tail criterion for strong convergence.

Solved by gpt-5.6-sol high.

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