Test Functions - Part 2

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Test Functions - Part 2
Prof. S Kesavan
Department of Mathematics
The Institute of Mathematical Sciences
  1. NPTEL-NOC IITM
  2. locally finite sets
  3. Existence of locally finite sets.
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Thanks for releasing these awesome lectures for free!

tanchienhao
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Something is off with the statement of existence of a locally finite smooth partition of unity. In the simplest case where one works with an open interval in R and the trivial open cover consisting of the interval itself, the family of smooth functions (indexed by the same set as the indexing set of the open cover) can consist of just one smooth function - but this is clearly not sufficient to simultaneously satisfy all the four conditions. The function must be identically 1 in the interval (in line with requirement 4), and therefore by smoothness (hypothesis) is also equal to 1 on the endpoints. But then its support is no longer contained in the interval (breaking requirement 1). I think this can be remedied if the indexing set for the family of smooth functions allows for redundancy (I haven't looked at a proof yet, nor have I worked out the details).

mmaannaann