A First Course in Functional Analysis by Orr Moshe Shalit

By Orr Moshe Shalit

Written as a textbook, A First path in practical Analysis is an advent to uncomplicated sensible research and operator thought, with an emphasis on Hilbert house tools. the purpose of this booklet is to introduce the fundamental notions of sensible research and operator concept with out requiring the coed to have taken a path in degree conception as a prerequisite. it really is written and based the way in which a direction will be designed, with an emphasis on readability and logical improvement along genuine functions in research. The historical past required for a scholar taking this direction is minimum; uncomplicated linear algebra, calculus as much as Riemann integration, and a few acquaintance with topological and metric spaces.

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15), thus the vector space operations and the inner product of G extend to H to make it a Hilbert space. Now to the details. Since V is injective, we can make V (G) into an inner product space by defining vector space operations cV (g) = V (cg), V (f ) + V (g) = V (f + g), and an inner product V (f ), V (g) = f, g , for f, g ∈ G and c ∈ C. To simplify notation, let us identify G with V (G), and omit V . We can do this because V is injective, and because V is compatible with the only structure that we initially had on V (G), which is the metric.

3. The space H3 = {f : D → C : f is bounded and analytic}. 4. The space H4 = f : D → C : f is analytic and ∞ k=0 |f (k) (0)|2 < ∞ . 5. Let [·, ·] : Cn × Cn → C. Prove that [·, ·] is an inner product on Cn if and only if there exists a positive definite matrix A ∈ Mn (C) such that [x, y] = Ax, y , where ·, · denotes the standard inner product on Cn . Likewise, characterize real-valued inner products on the real space Rn . 6. Define a norm · on R2 by (x1 , x2 ) = max{|x1 |, |x2 |}. Can you find an inner product on R2 such that · is the norm induced by this inner product?

10. When trying to think of L2 [a, b] as a space of functions, there are two questions to ponder. First, given an element f ∈ L2 [a, b], in what sense can it be considered as a function? We addressed this issue above. Second, given a function f : [a, b] → C, we ask whether or not f can be considered as an element in L2 [a, b]. This is an important question, but for our purposes it is not crucial to settle it now. 1. For a full understanding of L2 [a, b], the student is advised to learn the theory of Lebesgue measure and integration.

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