Uniform-norm of $f_n(x)=ne^{-nx^2}$

Question

Consider a sequence of functions, $(f_n(x))=ne^{-nx^2}$.

I think the uniform norm is $ne^{-n}$, but according to my solution, it is $n$.

Why is this the case? Don't we just take out the $x$ for the uniform norm of sequences of functions?

Answer 1

The uniform norm is usually defined by

$$\|f\| = \sup_{x \in \mathbb{R}} |f(x)|$$

Now for $x \ne 0$, $0 < e^{-nx^2} < 1$, so $f_n$ attains its maximum at $0$, and the maximum is $n$.