Further notes on Axler 5A

exercises/axler-5a.org

@@ -10,7 +10,28 @@ under $T$.
 
 ** Answer
 
-Let $U_\lambda$ ($\lambda\in\Lambda$) be some family of subspaces such that each $U_\lambda$
-is invariant under $T$. 
+First, what is a "collection of subspaces"? A collection of subspaces
+is:
+- A set, $\Lambda$, whose elements will be used to "index" the collection;
+- For each $\lambda\in\Lambda$, a subspace $U_\lambda$.
+
+We say that $U_\lambda$ (where $\lambda\in \Lambda$) is a "family of subspaces indexed by
+$\Lambda$".
+
+(You might think that we could have written $U_1$, $U_2$, ... and
+so on, using the natural numbers to index the~$U$s. But the question
+said /every/ collection, and some collections have "more things in
+them than there are natural numbers." Anyway, none of this is going to
+be very critical.)
+
+So, let $U_\lambda$ ($\lambda\in\Lambda$) be some family of subspaces such that each $U_\lambda$
+is invariant under $T$. What that means is that, for any $\lambda$ and for
+any $w\in U_\lambda$, we have $Tw\in U_\lambda$.
+
+The intersection of all the $U_\lambda$ is
+\begin{equation}
+\bigcap_{\lambda\in\Lambda} U_\lambda = \{ u \mid u\in\U_\lambda \text{ for every }\lambda\}.
+\end{equation}
+