Let V be a real or complex vector space (possibly infinite-dimensional), and let 3 be a
Chapter 6, Problem 22(choose chapter or problem)
Let V be a real or complex vector space (possibly infinite-dimensional), and let 3 be a basis for V. For x.y E V there exist v\,v2.... ,v E ft such that Define x = 2_]a i v i an( i y = /^hVjI I i=\ fay) = ])2"i.bi. (a) Prove that ( ) is an inner product, on V and that 0 is ail orthonormal basis for V. Thus every real or complex vector space; may be regarded as an inner product space. (b) Prove that if V R" or V = C" and ft is the standard ordered basis, then the inner product, defined above is the standard inner product.
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