$x \mapsto \abs{x_1} + \abs{x_2}$ on $\reals^2$

Example · Undergraduate

Terms

Term	This example	Why
norm	✓	$x \mapsto \abs{x_1} + \abs{x_2}$ is a norm.
inner product	✗	$x \mapsto \abs{x_1} + \abs{x_2}$ does not come from an inner product, because it fails the parallelogram law: with $x = (1, 0)$ and $y = (0, 1)$, $\norm{x + y}^2 + \norm{x - y}^2 = 8$, but $2 \norm{x}^2 + 2 \norm{y}^2 = 4$.

Distinctions using this example

• Norm vs. metric vs. inner product

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