MATH 304 Linear Algebra

MATH 304 Linear AlgebraLecture 20:Inner product notion ofnormgeneralizes the notion of lengthof a vector a vector space. A function :V Ris called anormonVif it has thefollowing properties:(i) (x) 0, (x) = 0 only forx=0(positivity)(ii) (rx) =|r| (x) for allr R(homogeneity)(iii) (x+y) (x) + (y) (triangle inequality) norm of a vectorx Vis usuallydenotedkxk. Different norms onVaredistinguished by subscripts, , V=Rn,x= (x1,x2, .. ,xn) Rn. kxk = max(|x1|,|x2|, .. ,|xn|). kxkp=(|x1|p+|x2|p+ +|xn|p)1/p,p V=C[a,b],f: [a,b] R. kfk = maxa x b|f(x)|. kfkp=( ba|f(x)|pdx)1/p,p vector vector spaceis a vectorspace endowed with a norm defines a distance function on the normedvector space:dist(x,y) =kx we say that a sequencex1,x2, ..convergesto a vectorxifdist(x,xn) 0 asn .Also, we say that a vectorxis a goodapproximationof a vectorx0ifdist(x,x0) is productThe notion ofinner productgeneralizes the notionof dot product of vectors a vector space.

Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in Rn. Definition. Let V be a vector space. ... where w is bounded, piecewise continuous, and w > 0 everywhere on [a,b]. w is called the weight function. Theorem Suppose hx,yi is an inner product on a vector ...

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