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Prove Minkowski's inequality using Cauchy-Schwarz's
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[QUOTE="logan3, post: 4963978, member: 453643"] [h2]Homework Statement [/h2] For [b]u[/b] and [b]v[/b] in [itex]R^n[/itex] prove Minkowski's inequality that [itex]\|u + v\| \leq \|u\| + \|v\|[/itex] using the [i]Cauchy-Schwarz inequality[/i] theorem: [itex]|u \cdot v| \leq \|u\| \|v\|[/itex]. [h2]Homework Equations[/h2] Dot product: [itex]u \cdot v = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n[/itex] Norm: [itex]\|u \| = \sqrt {u \cdot u}[/itex] Cauchy-Schwarz inequality: [itex]|u \cdot v| \leq \|u\| \|v\|[/itex] [h2]The Attempt at a Solution[/h2] Using def. of norm: [itex]\|u + v\|^2 = \sqrt {(u + v) \cdot (u + v)}^2 = (u + v) \cdot (u + v)[/itex] Expand: [itex](u + v) \cdot (u + v) = u \cdot u + 2 (u \cdot v) + v \cdot v[/itex] Using the Cauchy-Schwarz inequality: [itex]u \cdot u + 2 (u \cdot v) + v \cdot v \leq \|u \|^2 + 2 \|u \| \|v\| + \| v \|^2 = (\|u \| + \| v \|)^2[/itex] Therefore, [itex]\|u + v\|^2 \leq (\|u \| + \| v \|)^2 \Rightarrow \|u + v\| \leq \|u \| + \| v \|[/itex]. Thank-you [/QUOTE]
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Prove Minkowski's inequality using Cauchy-Schwarz's
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