By Krantz S.G.
This e-book is a short yet targeted and cautious creation to the topic of practical research. It covers the elemental themes that may be present in a easy graduate research textual content. however it additionally covers extra refined subject matters comparable to spectral thought, convexity, and fixed-point theorems. a different function of the e-book is that it features a nice many examples or even a few functions. It concludes with an announcement and evidence of Lomonosov's dramatic consequence approximately invariant subspaces
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Extra resources for A Guide to Functional Analysis
6. If E Â H is a closed subspace and E ¤ H then there is a y 2 H , y ¤ 0, such that y ? E. Proof. Take x 2 H , x 62 E. Set y D Qx. Since x ¤ P x, it follows that y ¤ 0. We next treat one of the most central and significant results of Hilbert space theory. Often called the Riesz representation theorem, it says that every bounded linear functional on a Hilbert space H is given by inner product with an element of H . 7. If is a bounded linear functional on the Hilbert space H , then there is a unique element y 2 H such that x D hx; yi for all x 2 H .
2. The Geometry of Hilbert Space 41 The linearity of shows that E is a subspace, and the fact that is continuous shows that E is closed. Since h ¤ 0 for some h 2 H , we know that E ? does not consist of 0 alone. Clearly the y that we seek lives in E ? Also y D hy; yi. Choose a nonzero z 2 E ? Then z 62 E, so z ¤ 0. Set y D ˛z, where ˛ D . z/=hz; zi. Then y 2 E ? , y D hy; yi, and y ¤ 0. For any x 2 H , we put x0 D x x y hy; yi and x 00 D x y: hy; yi Then x 0 D 0, hence x 0 2 E. Thus hx 0 ; yi D 0.
1 Introduction If we place some additional structure on a Banach space, then a richer theory results. That is what we now do. The key idea is to equip the Banach space with an inner product. That gives us notions of orthogonality and projection, and the result is a beautiful and coherent theory. D EFINITION . A complex vector space H is called an inner product space if, to each ordered pair of vectors x; y 2 H , there is associated a complex number hx; yi. This number is called the inner product or scalar product of x and y.
A Guide to Functional Analysis by Krantz S.G.