3.60A Euclidean space is a finite-dimensional normed space, which is complete (Propo-sition 1.4).3.61𝑓(x,y) =x𝑇ysatisfies the requirements of Exercise 3.59 and therefore(x𝑇y)2≤(x𝑇x)(y𝑇y)Taking square rootsx𝑇y≤ ∥x∥ ∥y∥3.62By definition, the inner product is a bilinear functional. To show that it is contin-uous, let𝑋be an inner product space with inner product denote byx𝑇y. Letx𝑛→xandy𝑛→ybe sequences in𝑋.(x𝑛)𝑇y𝑛−x𝑇y= (x𝑛)𝑇y𝑛−(x𝑛)𝑇y+ (x𝑛)𝑇y−x𝑇y≤(x𝑛)𝑇y𝑛−(x𝑛)𝑇y+ (x𝑛)𝑇y−x𝑇y≤(x𝑛)𝑇(y𝑛−y) + (x𝑛−x)𝑇yApplying the Cauchy-Schwartz inequality(x𝑛)𝑇y𝑛−x𝑇y≤ ∥x𝑛∥ ∥y𝑛−y∥+∥x𝑛−x∥ ∥y∥Since the sequencex𝑛converges, it is bounded, that is there exists𝑀such that∥x𝑛∥ ≤𝑀for every𝑛. Therefore(x𝑛)𝑇y𝑛−x𝑇y≤ ∥x𝑛∥ ∥y𝑛−y∥+∥x𝑛−x∥ ∥y∥ ≤𝑀∥y𝑛−y∥+∥x𝑛−x∥ ∥y∥ →03.63Applying the properties of the inner product∙ ∥x∥=√x𝑇x≥0∙ ∥
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Hilbert space, Inner product space, Michael Carter, Foundations of Mathematical Economics