A First Course in Functional Analysis by Orr Moshe Shalit

By Orr Moshe Shalit

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

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This is the geometric meaning of the condition Re h − g, f − g ≤ 0 above. Proof. Let g = PS (h) and f ∈ S. Expanding the inequality ≥ h−g 2 − 2 Re h − g, t(f − g) + t(f − g) 2 h − (tf + (1 − t)g) 2 (which holds for all t ∈ (0, 1)), we get h−g 2 ≥ h − g 2. Dividing by t and cancelling some terms, we obtain 2 Re h − g, f − g ≤ t f − g 2 . Since this is true for all t ∈ (0, 1), we conclude that 2 Re h − g, f − g ≤ 0. To get the converse implication, let f ∈ S. Then using 2 Re h−g, f −g ≤ 0 we find h−f 2 = (h − g) − (f − g) = h−g 2 − 2 Re h − g, f − g + f − g ≥ h − g 2, so g = PS (h).

To show that H is a Hilbert space, we need to show that the inner product induces the metric of H. Letting h, k, fn , gn as above, we have h−k 2 H = h − k, h − k H = lim n→∞ fn − gn 2 G. But since the metric in any metric space is continuous, we also have dH (h, k) = lim dG (fn , gn ) = lim n→∞ n→∞ fn − gn . Therefore the inner product on H induces a complete metric, as required. We now come to the final assertion. Let (H ′ , V ′ ) be another pair consisting of a Hilbert space and a linear map V ′ : G → H ′ as described in the theorem.

To prove the uniqueness, let g, h ∈ S such that g = h = d. If we form the sequence g, h, g, h, . , then we have just seen above that this is a Cauchy sequence. It follows that g = h. 3. Let S be a closed convex set in a Hilbert space H, and let h ∈ H. Then there exists a unique g ∈ S such that g−h ≤ f −h for all f ∈ S. Orthogonality, projections, and bases 31 Proof. 2 to the convex set S − h = {g − h : g ∈ S}. The unique element y ∈ S − h of minimal norm corresponds to a unique g ∈ S (given by g = y + h) such that g − h is minimal, so the theorem follows.

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