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Unformatted text preview: Math Camp 1: Functional analysis About the primer Goal To brieﬂy review concepts in functional analysis that ∗ will be used throughout the course. The following concepts will be described 1. Function spaces 2. Metric spaces 3. Dense subsets 4. Linear spaces 5. Linear functionals ∗ The definitions and concepts come primarily from “Introductory Real Analysis” by Kolmogorov and Fomin (highly recommended). 6. Norms and seminorms of linear spaces 7. Euclidean spaces 8. Orthogonality and bases 9. Separable spaces 10. Complete metric spaces 11. Hilbert spaces 12. Riesz representation theorem 13. Convex functions 14. Lagrange multipliers Function space A function space is a space made of functions. Each function in the space can be thought of as a point. Ex amples: 1. C [ a, b ], the set of all realvalued continuous functions in the interval [ a, b ]; 2. L 1 [ a, b ], the set of all realvalued functions whose ab solute value is integrable in the interval [ a, b ]; 3. L 2 [ a, b ], the set of all realvalued functions square inte grable in the interval [ a, b ] Note that the functions in 2 and 3 are not necessarily continuous! Metric space By a metric space is meant a pair ( X, ρ ) consisting of a space X and a distance ρ , a singlevalued, nonnegative, real function ρ ( x, y ) defined for all x, y ∈ X which has the following three properties: 1. ρ ( x, y ) = 0 iff x = y ; 2. ρ ( x, y ) = ρ ( y, x ); 3. Triangle inequality: ρ ( x, z ) ≤ ρ ( x, y ) + ρ ( y, z ) Examples 1. The set of all real numbers with distance ρ ( x, y ) =  x − y  is the metric space IR 1 . 2. The set of all ordered ntuples x = ( x 1 , ..., x n ) of real numbers with distance n ρ ( x, y ) = ( x i − y i ) 2 i =1 is the metric space IR n . 3. The set of all functions satisfying the criteria f 2 ( x ) dx < ∞ with distance ρ ( f 1 ( x ) , f 2 ( x )) = ( f 1 ( x ) − f 2 ( x )) 2 dx is the metric space L 2 (IR). 4. The set of all probability densities with KullbackLeibler divergence p 1 ( x ) ρ ( p 1 ( x ) , p 2 ( x )) = ln p 1 ( x ) dx p 2 ( x ) is not a metric space. The divergence is not symmetric ρ ( p 1 ( x ) , p 2 ( x )) = ρ ( p 2 ( x ) , p 1 ( x )) . Dense A point x ∈ IR is called a contact point of a set A ∈ IR if every ball centered at x contains at least one point of A . The set of all contact points of a set A denoted by ¯ A is called the closure of A . Let A and B be subspaces of a metric space IR. A is said to be dense in B if B ⊂ ¯ A . In particular A is said to be everywhere dense in IR if ¯ = R . A Examples 1. The set of all rational points is dense in the real line. 2. The set of all polynomials with rational coeﬃcients is dense in C [ a, b ]....
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.
 Spring '04
 RuthRosenholtz

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