Suppose (e1,e2,...,en) is an orthonormal basis of the inner product space V and v1,v2,...,vn are vectors of V such that ||ej−vj||<1/√n. Prove that (v1,v2,...,vn) is a basis of V.
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Expert's answer
2021-07-28T17:37:34-0400
Note: It is enough to show that (v1…,vn) are linearly independent.Suppose vi are linearly dependent then there are scalars (λ1,…,λn) whic are not all zero such that ∑i=1nλivi=0Then 0=∑i=1nλivi=∑i=1nλi(vi−ei)+∑i=1nλiei⟹∥∑i=1nλi(vi−ei)∥=∥∑i=1nλiei∥However, ∥∑i=1nλiei∥=∥∑i=1nλi(vi−ei)∥≤∑i=1n∣λi∣∥vi−ei∥<n1∑i=1n∣λi∣≤(∑i=1n∣λi∣2)21(Cauchy-Schwarz)⟹∥∑i=1nλiei∥≤(∑i=1n∣λi∣2)21(1)Since ei are orthonormal we have ∥∑i=1nλiei∥2=∑i=1n∣λi∣2So we have: ∥∑i=1nλiei∥=(∑i=1n∣λi∣2)21which is a contradiction to what we have above (i.e inequality (1)) hence showing that (v1,…,vn) are linearly indepemdent and we have our result.
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