I am reading an excerpt from Infinite Dimensional Analysis by Aliprantis and on page 235 it claims that if $X$ is a normed space, then "$x$ is a weak* continuous linear functional by definition".
From my understanding, each $x\in X$ should define a linear functional in $X^{*}$ by $$\varphi_{x}(x^{*}) = \langle x, x^{*}\rangle$$ but I have no idea why the inverse image of an open set in $\mathbb{F}$ should be open in the weak* topology merely by definition or if this is the correct function to consider.
I take the definition of the weak star topology to be defined by the seminorms $\{p_{x}: x\in X\}$ where $p_{x}(x^{*}) = \lvert \langle x, x^{*}\rangle\rvert$. Am I missing an obvious fact about $\varphi_{x}$ or something else that makes this obvious?