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Unformatted text preview: Nonlinear Systems and Control Lecture # 0 Mathematical Preliminaries – p.1/39 Vector Space (or Linear Space): A vector space ( V , F ) consists of a set (of vectors) V , a field (of scalars) F and two operations viz. addition of vectors (+) and multiplication of vectors by scalars ( · ) , which obey the following axioms: 1. Addition is given by + : V × V → V : ( x + y ) → x + y ; It is Associative: ( x + y ) + z = x + ( y + z ) , ∀ x,y,z ∈ V Commutative: x + y = y + x, ∀ x,y ∈ V ∃ ! identity , (called the zero vector), s.t. x + 0 = 0 + x = x, ∀ x ∈ V ∃ ! inverse: ∀ x ∈ V , ∃ !( − x ) ∈ V s.t. x + ( − x ) = 0 ; – p.2/39 2. Multiplication by scalars is given by · : F × V → V ( α, x ) → αx , where ∀ x ∈ V and ∀ α,β ∈ F , ( αβ ) x = α ( βx ) 1 x = x , x = 0 ; 3. Addition and multiplication by scalars are related by distributed laws viz. ∀ x ∈ V , ∀ α,β ∈ F ( α + β ) x = αx + βx ∀ x,y ∈ V , ∀ α ∈ F α ( x + y ) = αx + αy . Subspace: Let W be a subset of V . If W is a vector space itself, with the same vector space operations as V has, then it is a subspace of V . – p.3/39 Inner Product Space: A vector space ( V , F ) is an inner product space if there is a function (· , ·) : V × V mapsto→ F such that for every x,y,z ∈ V and α ∈ F the following hold: ( x + y,z ) = ( x, z ) + ( y,z ) , ( x,y + z ) = ( x, y ) + ( x,z ) ( x,αy ) = α ( x,y ) ∀ α ∈ F bardbl x bardbl 2 = ( x,x ) ≥ ( x,x ) = 0 if and only x = 0 . ( x,y ) = ( y,x ) , where the overbar denotes the complex conjugate operator. The function (· , ·) is called the inner product on V . – p.4/39 The supremum or least upper bound (LUB) of a set S of real numbers is denoted by sup( S ) and is defined to be the smallest real number that is greater than or equal to every number in S . Every nonempty subset of the set of real numbers that is bounded above has a supremum that is also an element of the set of real numbers. sup { x ∈ R : 0 < x < 1 } = sup { x ∈ R : 0 ≤ x ≤ 1 } = 1 . The infimum or greatest lower bound (GLB) of a set S of real numbers is denoted by inf( S ) and is defined to be the biggest real number that is smaller than or equal to every number in S . Any bounded nonempty subset of the real numbers has an infimum in the nonextended real numbers. inf { x ∈ R : 0 < x < 1 } = 0 . – p.5/39 Vector Norms: Let X be a vector space, a realvalued function bardbl · bardbl defined on X is said to be a norm on X if it satisfies the following properties: bardbl x bardbl ≥ (positivity) ; bardbl x bardbl = 0 if and only if x = 0 (positive definiteness) ; bardbl αx bardbl =  α bardbl x bardbl , for any scalar α (homogeneity) ; bardbl x + y bardbl negationslash = bardbl x bardbl + bardbl y bardbl (triangle inequality) for any x ∈ X and y ∈ X ....
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 Spring '08
 CHOI
 Topology, Continuous function, Metric space, Compact space, Banach space

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