Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a function sequence $\{f_{n}\}$ in $X$ converging to $0$, such that $\{a_{n}\,f_{n}\}$ does not converge to $0$ for all sequences of scalars $\{a_{n}\}$ converging to $\infty$.
Hi everybody. This problem supports the point that the metrizability is necessary for $X$ to make this theorem true:
If a sequence of functions $\{f_{n}\}$ in a metrizable topological vector space $X$ converges to $0$, then there is a scalar sequence $\{a_{n}\}$ converging to $\infty$ such that $\{a_{n}\,f_{n}\}$ converges to $0$.
But I got no clue how to find that sequence like above. Also, topologize a vector space by family of seminorms makes me confused so much. So I hope someone can help me. Thanks.
I have an idea, I hope you can fill the gaps I leave.
Let $\mathcal{A}=\{\mathbf{a}=(a_n)\mid a_n\in\mathbb{N}_+, a_n\to\infty \text{ and } a_n\le a_{n+1}\}$ and define a bijection between $[0,1]$ and $\mathcal{A}$, and denote the succession of $x\in[0,1]$ by $\mathbf{a}(x)$. For every $n$ define a function $f_n$ such that $f_n(x)>1/a_n(x)$, and take the functions such that $f_n\ge f_{n+1}>0$, then $\{f_n\}$ converge to a continuous function and try to arrange them, so they converge to zero in the rationals numbers (and so in all of $[0,1]$), so $f_n\to 0$ in the topology of $X$.
For every $\gamma_n\to\infty$ you can pick an $x$ such that we have $\{\gamma_nf_n(x)\}$ does not converge to zero and we are done.
I hope the idea is not too bad.