Topological Vectorspaces - ( August 30, 2005 ) Topological...

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Unformatted text preview: ( August 30, 2005 ) Topological vectorspaces Paul Garrett garrett@math.umn.edu http: / /www.math.umn.edu/˜garrett/ • A natural non-Fr´ echet space of functions • First definitions • Quotients and linear maps • More topological features • Finite-dimensional spaces • Baire category theorem • Seminorms and Minkowski functionals • Local convexity and Hahn-Banach theorems • Countably normed, countably Hilbert spaces • Local countability This little part is the first introduction to the notion of topological vectorspace in greatest generality. This would only be motivated after one is already acquainted with Hilbert spaces, Banach spaces, Fr´ echet spaces, and perhaps to understand that other important examples don’t fall into these classes. Some basic concepts are introduced which do not require the presence of a metric. Further, some concepts which would appear to depend upon having a metric are given sense in this more general context. The last point is that even in this generality finite-dimensional topological vectorspaces have just one possible topology. This has immediate consequences for maps to and from finite-dimensional topological vectorspaces. All this set-up works perfectly well with very mild hypotheses on the scalars involved. After this set-up, we are ready to look at the Baire Category Theorem and pursuant results: Banach- Steinhaus (Uniform Boundedness), Open Mapping, Closed Graph, etc. 1. A natural non-Fr´ echet space of functions There are many natural spaces of functions that are most definitely not Fr´ echet spaces. Most often these spaces have some sort of support condition. Let C o c ( R ) = { compactly-supported continuous C-valued functions on R } This is a strictly smaller space than the space C o ( R ) of all continuous functions on R , which we saw was Fr´ echet. Of course, we can express this function space as an ascending union C o c ( R ) = ∞ [ N =1 { f ∈ C o c ( R ) : spt f ⊂ [- N, N ] } Note, too, that each space C o N = { f ∈ C o c ( R ) : spt f ⊂ [- N, N ] } ⊂ C o [- N, N ] is strictly smaller than the space C o [- N, N ] of all continuous functions on the interval [- N, N ], since functions in C o N must have values 0 at the endpoints ± N . Still, C o N is a closed subspace of the Banach space C o [- N, N ] (with sup norm), since a sup-norm limit of functions vanishing at ± N must also vanish there. Thus, each individual C o N is a Banach space. Also, observe that for 0 < M < N the space C o M is a closed subspace of C o N (with sup norm), since the property of vanishing off [- M, M ] is preserved under sup-norm limits. 1 Paul Garrett: Topological vectorspaces (August 30, 2005) But for 0 < M < N the space C o M is nowhere dense in C o N , since an open ball of radius ε > 0 around any function in C o N contains many functions with non-zero values off [- M, M ]....
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Topological Vectorspaces - ( August 30, 2005 ) Topological...

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