Let x1, X2, X3, .., Xn, ... be any (countable set) of linearly independent elements of...

 

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Let x1, X2, X3, .., Xn, ... be any (countable set) of linearly independent elements of an inner product space V. Then V contains a set e,, e2, e3, ..., en, .. such that (a) The system in e, e2, e3, .., en, .. is orthonormal. (b) The space generated by x,,x2, X3, --, Xn ... and e,,e2, e3, ... , en, ... are the same. Let x1, X2, X3, .., Xn, ... be any (countable set) of linearly independent elements of an inner product space V. Then V contains a set e,, e2, e3, ..., en, .. such that (a) The system in e, e2, e3, .., en, .. is orthonormal. (b) The space generated by x,,x2, X3, --, Xn ... and e,,e2, e3, ... , en, ... are the same.