By Daniel Alpay

This is an workouts e-book before everything graduate point, whose target is to demonstrate a number of the connections among practical research and the idea of capabilities of 1 variable. A key function is performed via the notions of optimistic sure kernel and of reproducing kernel Hilbert house. a few evidence from sensible research and topological vector areas are surveyed. Then, numerous Hilbert areas of analytic features are studied.

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**Extra info for An Advanced Complex Analysis Problem Book: Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions**

**Example text**

III-13 and III-14]. 5. 3. Let M1 , . . , MN be a family of commuting n × n Hermitian matrices with complex entries. Show that Cn has a basis made of common eigenvectors of M1 , . . , MN . 4. Let M1 , M2 , . . , be a (not necessarily ﬁnite) family of commuting n × n matrices. (1) Show that they have a common invariant subspace. (2) Show that they have a common eigenvector. (3) Show that they can be simultaneously triangularized. In the statement of the next theorem, C is a smooth Jordan curve.

VN ) ⊂ C1 , the result follows. That the converse is not true is seen by taking the ﬁeld of rational functions in one variable. 5: Following for instance [225] we proceed by induction and consider ﬁrst the case N = 1. In the arguments it is well to recall that the quotient of a commutative ring by a maximal ideal is a ﬁeld. Case N = 1: The space Iz1 = {f ∈ C2 [X] ; f (z1 ) = 0 } 48 Chapter 1. Algebraic Prerequisites is a maximal ideal, and its quotient ﬁeld is C2 [z1 ]. This last space is a ﬁnitedimensional vector space over C2 since z1 is algebraic over C2 .

ZN ]. 1: Let N be the required space, and let M be a subspace both invariant under A and B and on which they coincide. Then, M ⊂ ker(A − B). Since AM ⊂ M we have, for m ∈ M, A(Am) = B(Am), and so, since Am = Bm, A(Am) = B(Bm), and so m ∈ ker(A2 − B 2 ). Thus the required space is included in u u ∩∞ u=1 ker(A − B ). This space is invariant under A and B, and A = B on it. Thus u u N = ∩∞ u=1 ker(A − B ). 1. Since U → ∩U u=1 ker(A − B ) is a decreasing sequence of linear n subspaces in C there exists an integer m such that u u N = ∩m u=1 ker(A − B ).