Interiors of sets in topological vector spaces

  • #1
In Rudin's book Functional Analysis, he makes the following claim about the interior [itex]A^\circ[/itex] of a subset [itex]A[/itex] of a topological vector space [itex]X[/itex]: If [itex]0 < |\alpha| \leq 1[/itex] for [itex]\alpha \in \mathbb C[/itex], it follows that

[tex]
\alpha A^\circ = (\alpha A)^\circ,
[/tex]

since scalar multiplicaiton (the mapping [itex]f_\alpha: X \to X[/itex] given by [itex]f_\alpha(x) = \alpha x[/itex]) is a homeomorphism; that is, a 1-to-1 continuous mapping.

This is the first time I've ever confronted the term "homeomorphism," and I have no idea how I can deduce the above observation about interiors given this hypothesis. Can someone help? (This is not a HW problem; it's a point in our notes that is confusing me.)
 

Answers and Replies

  • #2
Landau
Science Advisor
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You haven't seen the definition of homeomorphism? Its the topological version of 'isomorphism': an invertible continuous function with continuous inverse; this must be in Rudin also. Note: you say "a 1-to-1 continuous mapping", this is wrong. I hope by 1-1 you mean 'bijective' (I find this a confusing term, I have seen people use it for 'injective'), but the inverse is also required to be continuous!
 

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