An example of a norm which can't be generated by an inner product

Question

I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can generate.

Answer 1

For example, any $p$-norm except the $2$-norm.

To check this, any norm obtained from the inner-product should satisfy the parallelogram law. Whereas the $p$-norm with $p \neq 2$, does not satisfy the parallelogram law.

Answer 2

For example, the supremum norm $\lVert f \rVert_\infty = \max\{ \lvert f(t) \rvert : t \in [a,b]\}$ on $C[a,b]$ is not induced by an inner product. Take $f(t) = 1$ and $g(t) = \frac{t-a}{b-a}$ with $g(a) = 0$ and $g(b) = 1$, so that $$ \lVert f + g \rVert_\infty = 2, \quad \lVert f - g \rVert_\infty = 1. $$ Then $2 \lVert f \rVert_\infty^2 + 2 \lVert g \rVert_\infty^2 = 4$, but $\lVert f + g \rVert_\infty^2 + \lVert f - g \rVert_\infty^2 = 5$, thus violating the parallelogram law.

In addition, one can also verify that $f(t) = \max\{\sin(t),0\}$ and $g(t) = \max\{-\sin(t),0\}$ are counterexamples on $(C[0,2 \pi], \lVert\;\cdot\;\rVert_\infty)$.