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Unformatted text preview: Lecture 1 Definition 1. A norm on the real vector space V is a map k • k : V → R with the following properties (for t ∈ R and x,y,z ∈ V ): k z k > and k z k = 0 ⇔ z = 0; k tz k = | t |k z k ; k x + y k 6 k x k + k y k . The rule x,y ∈ V ⇒ d( x,y ) = k x- y k then defines a metric d on V ; this is an easy exercise . All subsequent metric properties of a normed space refer to this. Definition 2. A Banach space is a complete normed space. This is the first such metric reference; completeness is of course in the Cauchy sense. Definition 3. An inner product on the real vector space V is a real-bilinear map ( •|• ) : V × V → R that is symmetric: x,y ∈ V ⇒ ( x | y ) = ( y | x ) and positive-definite: 6 = z ∈ V ⇒ ( z | z ) > . 1 The rule z ∈ V ⇒ k z k = p ( z | z ) then defines a norm on V ; a standard proof of this rests on the following Cauchy-Schwarz inequality....
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