Assume that a continuous function $f(x)$ has uniform norm 5 on the
interval [1,4]. What is the largest possible value of the $L_2$-norm of
$f(x)$ on the intervals [1,4] and on [2,4]?
From what I understand about norms, the $L_2$ norm is greater than the $L_\infty$ norm...and I don't see a bound of sorts. How would I deduce the answers?
Answer
Hint: if the uniform norm of f is 5 on [1,4] then $$\sup_{x\in [1,4]} |f(x)|=5$$
$$\int_{1}^{4}|f(x)|^2 dx \leq \int_{1}^{4} 5^2 dx $$
$$\int_{2}^{4}|f(x)|^2 dx \leq \int_{2}^{4} 5^2 dx $$
I think you might have been confused by the fact that "$L_2$ is greater than the uniform norm". In fact it is true that some norms are greater than others but they can be bounded by a smaller norm up to a constant. For instance in $\mathbb{R^n}$ in general $L_1\geq L_{\infty}$ but $L_1\leq n L_{\infty}$.
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