 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2011/1/solution

R

X

∥ \cdot ∥ : X \to [0, \infty)

satisfying

∥ x ∥ = 0 ⟺ x = 0, ∥ a x ∥ = ∣ a ∣∥ x ∥, ∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥.

d (x, y) = ∥ x - y ∥

. It is a Banach space if this metric is complete: every Cauchy sequence converges to an element of $X$ .

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