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Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · Analysis & Linear Algebra › Metric Spaces

“d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point” — false

Counterexample: T(x) = x + 1/x on [1, ∞)

Distances strictly decrease but no contraction constant k < 1 exists; no fixed point.

completenessfixed point