## Show that if $|f(x)| \leq \phi(x) + \psi (x)$, exist $g,h$ such that $f(x) = g(x)+h(x),\, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x)$

Given $$\phi$$ and $$\psi$$ two seminorms in a vector space X, and a functional $$f:X \rightarrow \mathbb{K}$$, where $$\mathbb{K} = \mathbb{R}$$ or $$\mathbb{C}$$, such that $$|f(x)| \leq \phi(x) + \psi (x) \, \, \forall x \in X$$, show that there exist two linear functionals $$g,h: X \rightarrow \mathbb{K}$$ such that: $$f(x) = g(x) + h(x), \, \, \, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x) \, \, \, \forall x \in X$$.

It is obvious that if $$g,h$$ exist, with the indicated bound, then $$|f(x)| \leq \phi(x) + \psi (x)$$, but how can I show that these functionals exist?