Analyse Complexe by Eric Amar, R. Gay, T. V. Nguyen

By Eric Amar, R. Gay, T. V. Nguyen

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Let g : E → R be a bounded continuous function. Then there exists a bounded continuous function f : X → R so that f E = g. 1) Tx Y ≤ C x X The smallest C defines a norm, T , on L(X, Y ), the bounded linear maps of X to Y . L(X, Y ) with this norm is a Banach space if and only if Y is a Banach space. L(X) ≡ L(X, X) is the major object of study in this volume. We use K for R or C, the field over which an NLS is a vector space. L(X, K) ≡ X ∗ is the dual space for X. 4 of Part 1). Occasionally, we will use ideas from the theory of locally convex spaces and tempered distributions (Chapter 6 of Part 1) and our examples often use the theory of Fourier transforms (also Chapter 6 of Part 1).

Preliminaries If πn ∈ Σn is a permutation of {1, . . , n}, then we define σπ : ⊗n V → ⊗n V by σπ (B)( 1 , . . , n ) = B( π−1 (1) , . . 37) Recall each permutation π ∈ Σn has a sign, (−1)π ∈ {−1, 1} defined in Problem 3. 38) If we define on ⊗n V , An = it is exactly the projection onto ∧n (V ) to itself, and we use 1 n! 39) ⊂ ⊗n V . 33), Given f1 , . . , fn ∈ V, define f1 ∧ · · · ∧ fn ∈ ∧n (V ) by √ f1 ∧ · · · ∧ fn ≡ n! 44) =√ n! π∈Σn √ Where we use n! ) and others n! ). It passes through all the calculations below.

Xπ(n) ) = (−1)π P (x1 , . . , xn ) (b) Prove that (−1)ππ = (−1)π (−1)π . Licensed to AMS. 3. Some Linear Algebra 19 (c) Let i = j. 79) (d) If (i1 , i2 , . . , i ) (for all unequal i’s) is the permutation ⎧ ⎪ k∈ / {i1 , . . , i } ⎨k, (i1 , . . , i ) = ij+1 , k = ij , j = 1, 2, . . 81) (Hint: Prove that (i1 , . . ) 4. Let N ∈ L(V ) with N = 0. Prove that Ker(N ) = {0}. (Hint: By decreasing , suppose N −1 = 0. ) 5. Let N be a nilpotent operator on a finite-dimensional vector space, X. Suppose m is such that N m = 0 but N m−1 = 0 (although N m−1 x may be zero for some x, just not for all x).

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