I am self learning Rudin's Functional Analysis Chp 6. I encountered a small problem regarding the topology of $D(\Omega)$.
Definitions Let $\Omega$ be a nonempty open set in $R^n$.
(a) For every compact $K\subset \Omega$, $\tau_k$ denotes the Frechet space topology of $D_k$.
(b) $\beta$ is the collection of all convex balanced sets $W\subset D(\Omega)$ such that $D_k\cap W \in \tau_k$ for every compact $K\subset\Omega$.
(c) $\tau$ is the collection of all unions of sets of the form $\phi+W$, with $\phi\in D(\Omega)$ and $W\in\beta$.
Theorem $\tau$ is a topology in $D(\Omega)$, and $\beta$ is a local base for $\tau$.
Suppose $V_1, V_2\in\tau,\phi\in V_1\cap V_2$. To prove it, it is clearly enough to show that $\phi + W\subset V_1\cap V_2$ for some $W\in\beta$.
The definition of $\tau$ shows that there exist $\phi_i\in D(\Omega)$ and $W_i\in\beta$ such that $\phi\in\phi_i +W_i\subset V_i\ (i=1,2)$. Choose $K$ so that $D_k$ contains $\phi_1,\phi_2$, and $\phi$. Since $D_k\cap W_i $ is open in $D_k$, we have $\phi - \phi_i\in(1-\delta_i)W_i$ for some $\delta>0$.The convexity of $W_i$ implies therefore that $\phi - \phi_i +\delta_i W_i\subset(1-\delta)W_i+\delta_i W_i = W_i$, so that $\phi +\delta_i W_i\subset \phi_i +W_i\subset V_i\ (i=1,2)$. Hence the desired result holds with $W=(\delta_1 W_1)\cap(\delta_1 W_2)$,and the theorem is proved.
I have several questions.
(a) Is it true that if $W\in\beta$, then $\delta W \in \beta$ for $\delta \leq 1$?
My guess: It is true. $\delta W$ is of course convex and balanced. For each $D_k$,$D_k\cap\delta W = \delta(1/\delta D_k\cap W)=\delta(D_k\cap W)$ since $D_k$ is a subspce. And in $\tau_k$, since it is induced from a norm, any multiple of an open set is still open. I have trouble in proving the very last bolded line.
(b) Why there must be some $\delta_i$ such that $\phi - \phi_i \in (1-\delta_i )W_i$?
I know that $\phi -\phi_i \in D_k\cap W_i$, which is open in $D_k$. But how to proceed.
Any help will be very appreciated!