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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Functional Analysis (Big Rudin)

**Physics Forums | Science Articles, Homework Help, Discussion**