## Primitive Recursion of two functions $g,h$

Which function is created when applying Primitive Recursion $$PR(g,h)$$ to \begin{align}g:\mathbb{N}\to \mathbb{N}, \quad g(n)&=\mathrm{zero}_1(n)\ h:\mathbb{N}^3\to\mathbb{N}, \quad h(n,m,l)&=P_2^{(3)}(n,m,l)+\mathrm{succ}\circ P_2^{(3)}(n,m,l)+P_3^{(3)}(n,m,l) ?\end{align}

My idea: $$f(x,0)=g(x)=zero_1(x)=0$$, $$f(x,t+1)=h(x,t,f(x,t))=t+t+1+f(x,t)=2t+1+f(x,t)$$, but what is $$f(x,t)$$ and how do I derive that?

## 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?