I had a quick question on a part of a proof in chapter 1 of Functional Analysis, by Professor Rudin.(adsbygoogle = window.adsbygoogle || []).push({});

Theorem 1.10 states

"Suppose K and C are subsets of a topological vector space X. K is compact, and C is closed, and the intersection of K and C is the empty set. Then 0 has a neighborhood V such that

[tex] (K+V) \cap (C+V) = \emptyset [/tex]"

In the proof of this theorem, Professor Rudin starts out by proving the following proposition

"If W is a neighborhood of 0 in X, then there is neighborhood U of 0 which is symmetric (in the sense that U = -U) and which satisfies

[tex] U + U \subset W [/tex]."

The question I have is about the next part of Rudin's proof

"Suppose K is not empty, and consider x in K, since C is closed, and since x is not in C, and since the topology of X is invariant under translations, the preceding proposition shows that 0 has a symmetric neighborhood

[tex] V_{x} [/tex]

such that

[tex] x + V_{x} + V_{x} + V_{x} [/tex]

does not intersect C..."

Is Professor Rudin's reasoning as follows:

Since C is closed, the the complement C* of C is open in X by definition. Since x is not contained in C, then x is contained in the complement of C, C*. Since C* is open, and contains x, then C* is a neighborhood of x. Since C* is a neighborhood of x, then the set

[tex]-x+C^{*}[/tex] is a neighborhood of 0 in X. Thus by the preceding proposition, there exists a symmetric neighborhood

[tex] V_{x}[/tex]

of 0 in X such that

[tex] V_{x} + V_{x} + V_{x} \subset -x + C^{*} [/tex].

Since the topology of X is translation invariant, then

[tex] V_{x} + V_{x} + V_{x} \subset -x + C^{*} [/tex] iff

[tex] x+V_{x} + V_{x} + V_{x} \subset x+(-x) + C^{*} = C^{*}[/tex],

so that [tex] x+V_{x} + V_{x} + V_{x}[/tex] does not intersect C...?

Is this line of reasoning correct?

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# Functional Analysis (Big Rudin)

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