to show cardinality in separable Hilbert space setting

Question

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the above exercise in Conway's Functional Analysis book

to me, the setting is too rough so i have no idea to step forward

could you help me to start this proof? or just cast a hint?

Answer 1

Answer for the first part: it is enough to show that $I_n \equiv \{i\in I: \frac 1 n \leq \mu (E_i) <\infty\}$ is at most countable for each positive integer $n$. Let $f_i=I_{E_i}$ for $i \in I_n$. Note that $\|f_i-f_j||_2\geq \sqrt {\frac 1 n+\frac 1 n}$ if $i \neq j$. Consider open balls around $f_i$ ($i\in I_n$) with radius $\frac 1 {\sqrt {2n}}$. These balls are disjoint. Take a countable dense subset of the space and pick an element form each of these balls. You get a one-to-one map from $I_n$ into the integers. Hence $I_n$ is countable.