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Fall 2015 | Homework 71 Solutions
This homework was due on Monday, September 21, in class.
1. Let H be a Hilbert space and M be a closed subspace of H. Suppose is a bounded
linear functional dened on M . Use Hilbert space methods to show that can be
uni
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Fall 2015 | Homework 72 Solutions
This homework was due on Wednesday, October 7, in class.
1. Let X and Y be Banach spaces and tTn u BpX, Y q. Show that the following are
equivalent:
(a) tTn u is bounded.
(b) tTn xu is bounded for all x P X.
(c) t|pTn x
Fall 2015 | Homework 73 | Due on Monday, November 9, in class
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1. If the inverse T 1 of a closed linear operator exists, show that T 1 is also a closed
linear operator.
2. Let T be a closed linear operator. If two sequences txn u and tn u in DpT q conver
Fall 2015 | Homework 72 | Due on Wednesday, October 7, in class
1
1. Let X and Y be Banach spaces and tTn u BpX, Y q. Show that the following are
equivalent:
(a) tTn u is bounded.
(b) tTn xu is bounded for all x P X.
(c) t|pTn xq|u is bounded for all x P
Fall 2015 | Homework 71 | Due on Monday, September 21, in class
1
1. Let H be a Hilbert space and M be a closed subspace of H. Suppose is a bounded
linear functional dened on M . Use Hilbert space methods to show that can be
uniquely extended to H, i.e.,