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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, ∀ μ ∈ Y * , ( Ax,μ ) = ( x,A * μ ) . (1.2) Further, show that bardbl A * μ bardbl ≤ bardbl A bardblbardbl μ bardbl , μ ∈ Y * . (1.3) Hint: Fix μ ∈ Y * . Then define A * μ : X → F so that (1.2) is satisfied, i.e., set ( x,A * μ ) = ( Ax,μ ) for x ∈ X . Show that A * μ defined in this way satisfies (1.3), and conclude that A * μ ∈ 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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