By Orr Moshe Shalit

Written as a textbook, **A First path in practical Analysis** is an advent to easy sensible research and operator concept, with an emphasis on Hilbert house tools. the purpose of this ebook is to introduce the elemental notions of useful research and operator concept with no requiring the coed to have taken a path in degree concept as a prerequisite. it's written and based the way in which a path will be designed, with an emphasis on readability and logical improvement along genuine purposes in research. The historical past required for a pupil taking this direction is minimum; easy linear algebra, calculus as much as Riemann integration, and a few acquaintance with topological and metric spaces.

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**Additional info for A First Course in Functional Analysis**

**Example text**

Proof. If f and g are linearly dependent, then without loss of generality f = cg, and | f, g | = |c| g 2 = f g . If f, g ∈ G are not linearly dependent (so in particular nonzero), then consider the unit vectors u = f1 f and v = 1g g. We need to prove that 16 A First Course in Functional Analysis | u, v | < 1. Let t ∈ C be such that |t| = 1 and t u, v = −| u, v |. Using the properties of inner product, we have 0 < u + tv 2 = u + tv, u + tv = u, u + t u, v + t v, u + |t|2 v, v = 2 − 2| u, v |. Thus | u, v | < 1 as required.

We are ready to start applying the structure theory of Hilbert spaces that we developed in the previous chapter, together with the Stone-Weierstrass theorem we proved in the introduction, to obtain some results on Fourier series. 1 An approximation result Recall that we defined L2 (K) to be the completion of the inner product space C(K) with respect to the inner product f, g = f (x)g(x)dx. K This ends up being the same space L2 (K) as one encounters in a course on Lebesgue measure and integration theory.

Xk ) ∈ K . The spaces P and Cper (K) are contained in C(K) and therefore also in L2 (K). We let · ∞ denote the sup norm in C(K), and we let · 2 denote the Hilbert space norm in L2 (K). 1. P is dense in Cper (K) in the norm. ∞ Proof. 3) from the introduction, together with the identities 2i sin t = eit − e−it and 2 cos t = eit + e−it . 7. 3, and identify Cper (K) with C(Tk ) via the map Φ : C(Tk ) → Cper (K) , Φ(f )(x) = f e2πix1 , . . , e2πixk . Here T = {z ∈ C | |z| = 1} and Tk = T × · · · × T (k times).