MIT18_102s09_lec01 - MIT OpenCourseWare

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Unformatted text preview: MIT OpenCourseWare 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: . 4 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 1. Tuesday, 3 Feb. Linear spaces, metric spaces, normed spaces. Banach spaces. Examples – Eu- clidean spaces, continuous functions on a closed interval – C ([0 , 1]) with supremum norm. The (Riemannian) L 1 norm, outline that this is not complete on C ([0 , 1]) . Brief description of l 2 – This is Hilbert space, but not explained. What is it all for? Main aims:- To be able to carry out ‘standard’ constructions in (linear) functional analysis: Abstract Hilbert space – one in each dimension Concrete Hilbert space – Many, such as L 2 ([0 , 1]) Example of a theorem:- The Dirichlet problem. Let V : [0 , 1] −→ R be a real-valued function. We are interested in ‘oscillating modes’ on the interval; something like this arises in quantum mechanics for instance. Namely we want to know about functions u ( x ) – twice continuously differentiable on [0 , 1] which satisfy the differential equation d 2 u (1.1) − dx 2 ( x ) + V ( x ) u ( x ) = λu ( x ) where λ is an ‘unknown’ constant – that is we want to know which λ ’s can occur. Well, of course all λ ’s can occur with u ≡ but this is the ‘trivial solution’ which will always be there for such an equation. What other solutions are there? Well, there is an infinite sequence of λ ’s for which there is a non- trivial solution of (1.1) λ j ∈ R – they are all real no non-real complex λ ’s can occur. For each of these there is at least one (and maybe more) ‘independent’ solution u j . We can say a lot more...
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This note was uploaded on 11/08/2011 for the course PHY 18.102 taught by Professor Staff during the Spring '09 term at MIT.

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MIT18_102s09_lec01 - MIT OpenCourseWare

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