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Prove Minkowski's inequality using Cauchy-Schwarz's

  1. Jan 3, 2015 #1
    1. The problem statement, all variables and given/known data
    For u and v in [itex]R^n[/itex] prove Minkowski's inequality that [itex]\|u + v\| \leq \|u\| + \|v\|[/itex] using the Cauchy-Schwarz inequality theorem: [itex]|u \cdot v| \leq \|u\| \|v\|[/itex].

    2. Relevant equations
    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]

    3. The attempt at a solution
    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
     
  2. jcsd
  3. Jan 3, 2015 #2

    jbunniii

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    Looks fine to me.
     
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