Answer to Question #272586 in Functional Analysis for Prathibha Rose

Question

Let x1 x2 … . . xn, be an orthonormal set in X and 𝑘1 ,𝑘2,. … … … 

𝑘nbe scalars

having absolute value 1. Then 𝑘1x1 + 𝑘 2x 2+ ⋯ … 𝑘 nxn= x1 + ⋯ + 

xn

Answer 1

a set of vectors is orthogonal if every pair of vectors is orthogonal

A set of vectors is orthonormal if it is an orthogonal set having the property that every vector is a unit vector 

since $k_n=1$ then:

$𝑘 _nx_n=x_n$

so, statement $𝑘_1x_1 + 𝑘 _2x_ 2+ ⋯ … 𝑘_ nx_n= x_1 + ⋯ + x_n$ is true