Let $X$ be a locally convex space whose topology is defined by a family of seminorms $\mathcal{P}$. Let $f$ be a linear functional on $X$. Then, I am trying to prove that the following statements are equivalent.
- $f$ is continuous.
- There are seminorms $p_1,...,p_n\in \mathcal{P}$ and positive scalars $\alpha_1,...,\alpha_n$ such that $|f(x)|\leq \sum_{k=1}^n\alpha_kp_k(x).$
I am stuck in proving $1\Rightarrow 2$. Applying the definition, and that $f$ is continuous at $0$, I get that there exist seminorms $p_1,...,p_n$ and $\epsilon_1,...,\epsilon_n>0$, such that whenever $x\in \bigcap_{k=1}^n\{y:p_k(y)<\epsilon_k\}\Rightarrow |f(x)|<1.$ But I am not able to conjure up positive $\alpha_1,..,\alpha_n$ such that above inequality holds. A hint will be very appreciated. Thanks.