Mathematics_1_oneside.pdf

Metric space the notion of a metric can be

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Metric space. The notion of a metric can be generalized. Let V be some Definition 8.15 vector space. Then any function d ( · , · ): V × V R that satisfies properties (i) d ( x , y ) = d ( y , x ) (ii) d ( x , y ) 0 where equality holds if and only if x = y (iii) d ( x , z ) d ( x , y ) + d ( y , z ) is called a metric . A vector space that is equipped with a metric is called a metric vector space . Definition 8.13 (and the proof of Theorem 8.14 ) shows us that any norm induces a metric. However, there also exist metrics that are not induced by some norm. Let L be the vector space of all random variables X on some given prob- Example 8.16 ability space with finite variance V ( X ). Then the following maps are metrics in L : d 2 : L × L [0, ), ( X , Y ) 7→ k X - Y k = p E (( X - Y ) 2 ) d E : L × L [0, ), ( X , Y ) 7→ d E ( X , Y ) = E ( | X - Y | ) d F : L × L [0, ), ( X , Y ) 7→ d F ( X , Y ) = max fl fl F X ( z ) - F Y ( z ) fl fl where F X denotes the cumulative distribution function of X .
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8.2 O RTHOGONALITY 59 8.2 Orthogonality Two vectors x and y are perpendicular if and only if the triangle shown below is isosceles, i.e., k x + y k = k x - y k . x - y y x + y x - y The difference between the two sides of this triangle can be computed by means of an inner product (see Problem 8.9 ) as k x + y k 2 -k x - y k 2 = 4 x 0 y . Two vectors x , y R n are called orthogonal to each other if x 0 y = 0. Definition 8.17 Pythagorean theorem. Let x , y R n be two vectors that are orthogonal Theorem 8.18 to each other. Then k x + y k 2 = k x k 2 +k y k 2 . P ROOF . See Problem 8.10 . Let v 1 ,..., v k be non-zero vectors. If these vectors are pairwise orthogo- Lemma 8.19 nal to each other, then they are linearly independent. P ROOF . Suppose v 1 ,..., v k are linearly dependent. Then w.l.o.g. there exist α 2 ,..., α k such that v 1 = k i = 2 α i v i . Then v 0 1 v 1 = v 0 1 ( k i = 2 α i v i ) = k i = 2 α i v 0 1 v i = 0, i.e., v 1 = 0 by Theorem 8.2 , a contradiction to our as- sumption that all vectors are non-zero. Orthonormal system. A set { v 1 ,..., v n } R n is called an orthonormal Definition 8.20 system if the following holds: (i) the vectors are mutually orthogonal, (ii) the vectors are normalized. Orthonormal basis. A basis { v 1 ,..., v n } R n is called an orthonormal Definition 8.21 basis if it forms an orthonormal system. Notice that we find for the elements of an orthonormal basis B = { v 1 ,..., v n } : v 0 i v j = δ i j = ( 1, if i = j , 0, if i 6= j .
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8.2 O RTHOGONALITY 60 Let B = { v 1 ,..., v n } be an orthonormal basis of R n . Then the coefficient Theorem 8.22 vector c ( x ) of some vector x R n with respect to B is given by c j ( x ) = v 0 j x . P ROOF . See Problem 8.11 . Orthogonal matrix. A square matrix U is called an orthogonal ma- Definition 8.23 trix if its columns form an orthonormal system. Let U be an n × n matrix. Then the following are equivalent: Theorem 8.24 (1) U is an orthogonal matrix. (2) U 0 is an orthogonal matrix. (3) U 0 U = I , i.e., U - 1 = U 0 . (4) The linear map defined by U is an isometry , i.e., k Ux k = k x k for all x R n .
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