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chap-4: Soit u : E −→ F linéaire, les assertions suivantes sontéquivalentes :(1) u est continue sur E,(2) ∃k ⩾ 0 : ∀x ∈ E, ∥u(x)∥F ⩽ k∥x∥E ,(3) u est lipschitzienne,(4) u est continue en 0E ,(5) u est bornée sur la sphère unité,(6) u est bornée sur la boule unité ouverte,(7) u est bornée sur la boule unité fermée.
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