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Unformatted text preview: FUNCTIONAL ANALYSIS LECTURE NOTES: ADJOINTS IN BANACH SPACES CHRISTOPHER HEIL 1. Adjoints in Banach Spaces If H , K are Hilbert spaces and A B ( H,K ), then we know that there exists an adjoint operator A * B ( K,H ), which is uniquely defined by the condition x H, y H, ( Ax,y ) K = ( x,A * y ) H . (1.1) Now we will consider the case where X , Y are Banach spaces and A B ( X,Y ). We will see that there exists a unique adjoint A * B ( Y * ,X * ) which is defined by an equation that generalizes equation (1.1) to the setting of Banach spaces. Note, however, that while we have A : X Y, the adjoint will be a map A * : Y * X * . In particular, unlike the Hilbert space case, we cannot consider compositions of A with A * . Exercise 1.1. Let X , Y be Banach spaces, and let A B ( X,Y ) be fixed. Show that there exists a unique operator A * B ( Y * ,X * ) that satisfies x X, Y * , ( Ax, ) = ( x,A * ) . (1.2) Further, show that bardbl A * bardbl bardbl A bardblbardbl bardbl , Y * . (1.3) Hint: Fix Y * . Then define A * : X F so that (1.2) is satisfied, i.e., set ( x,A * ) = ( Ax, ) for x X . Show that A * defined in this way satisfies (1.3), and conclude that A * X * , and also that A * is bounded. Finally, show there is no other map B : Y * X * that satisfies (1.2). The preceding exercise defines A * as a map from Y * to X * . Furthermore, equation (1.3) shows that bardbl A * bardbl bardbl A bardbl . The next exercise will show that equality holds. c circlecopyrt 2007 by Christopher Heil. 1 2 ADJOINTS IN BANACH SPACES Exercise 1.2. Let X , Y be Banach spaces, and choose A B ( X,Y ). This exercise will show that A * B ( Y * ,X * ) satisfies bardbl A * bardbl = bardbl A bardbl ....
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This note was uploaded on 08/25/2011 for the course MATH 6338 taught by Professor Staff during the Summer '08 term at Georgia Institute of Technology.
 Summer '08
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