Norm in $\mathcal{C}[\Bbb{R},\Bbb{R}]$

Question

I’m interested in expliciting a norm in the space of continuous functions from $\Bbb{R}$ to itself. It does not need to induce a complete metric.

Answer 1

Continuity implies Riemann integrability. Multiplication preserves continuity, so the product is Riemann integrable. For $a,b\in\mathbb{R}$ with $a<b$, we can define the inner product $\langle\cdot,\cdot\rangle: V\times V\to\mathbb{R}$ by $$ \langle f, g \rangle = \int_a^b fg $$ This induces a norm via $\langle f, f\rangle = \| f \|^2$.

EDIT: Only a semi-norm. Not an inner product space. Semi-norm is given by $$ \| f \| = \sqrt{\int_a^b f^2} $$ I believe you can quotient out by the kernal of the semi-norm to create a norm but this may be incorrect and probably violates the conditions of the question.