- #1

logan3

- 83

- 2

## Homework Statement

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].

## Homework 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]

## 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