1
$\begingroup$

Let $\{x_n\}_{n=1}^N$ be an orthonormal set in an inner product space $V$ with inner product $(\cdot,\cdot)$.

I am trying to show that $\left(\sum\limits_{n=1}^N(x_n,x)x_n,x-\sum\limits_{n=1}^N(x_n,x)x_n\right)=0.$

Reed & Simon, Functional Analysis, Theorem II.1 mention this is a “short computation using the properties of inner products” but I am having trouble with this.

  • 0
    Right. $x$ is an element of $V$.2012-12-09

1 Answers 1

1

$\left(\sum\limits_{n=1}^N(x_n,x)x_n\,\,,\,\,x-\sum\limits_{n=1}^N(x_n,x)x_n\right)=\sum_{n=1}^N(x_n,x)^2-\sum_{n=1}^N\sum_{k=1}^N(x_n,x)(x_k,x)(x_n,x_k)=$

$=\sum_{n=1}^N(x_n,x)^2-\sum_{n=1}^N(x_n,x)^2=0$

since $\,(x_n,x_k)=\delta_{n,k}\,$