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University of Toronto St. George
Sean Uppal

MAT224H1c.doc Inner Product Spaces INNER P RODUCTS Definition An inner producton a vector space V is a function that assigns a numberv,w to every pair v, w of vector space V in such a way that the following axioms holds: P 1 v,w is a real number. P 2: v,w = w,v . P 3 v+ w,u = v,u + w,u . P 4 rv,w = r v,w . P 5 v,v 0,vV . Definition A vector space V with an inner product is called an inner product space. Note , :V V R . V,R,+, ) is a vector space. V,R,+,, , ) is an inner product space. Examples The following are inner product spaces. 1) (R ,R,+,, , ), define X,Y = X Y the dot product. 2) C [a,b],R,+,, , , define f , g = bf(x ) x(dx). a T 3) (M mn,R,+,, , , define A, B = tr AB . Theorem Let , be an inner product on a space V. Let u, v, w denote vectors in V, r a real number. Then: 1) u,v + w = u,v + u, w . 2) v,rw = r v,w . 3) v,0 = 0 = 0,v . 4) v,v = 0 if and only if v = 0 . Theorem T n n If A is any nn positive definite matrix, thX,Y = X AY,X,Y R defines an inner product on R, and every inner product on R , and every inner product on R arises in this way. Proof: Page 1 of 9
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