Let $\{x_1,\dots,x_n\}$ be a orthonormal set. For $x\in H$ the vector $x-\sum_{k=1}^n \langle x,x_k\rangle x_k$ is orthogonal to $x_k, \forall k$

Question

Let $\{x_1,\dots,x_n\}$ be a finite orthonormal set in a Hilbert Space $H$. Prove that for any $x\in H$ the vector
$$x-\sum_{k=1}^n \langle x,x_k\rangle x_k$$
is orthogonal to $x_k$ for every $k=1,\dots,n$.

Since $\{ x_1,\dots,x_n\}$ is an orthonormal set, we know that $\langle x_i,x_j\rangle = 0$ for $i\neq j$. We will show that $\langle x-\sum_{k=1}^n \langle x,x_k\rangle x_k , x_k \rangle = 0$ for all $k=1,\dots,n$. Then,

\begin{align}
\langle x-\sum_{k=1}^n \langle x,x_k\rangle x_k , x_k \rangle &= \langle x,x_k\rangle -\langle \sum_{k=1}^n \langle x,x_k\rangle x_k , x_k\rangle \\
&= \langle x,x_k\rangle – \langle \langle x,x_k\rangle x_k, x_k \rangle \\
&= \langle x, x_k \rangle – \langle x,x_k\rangle \langle x_k,x_k\rangle \\
&= \langle x,x_k\rangle – \langle x,x_k\rangle \| x_k\|
\end{align}
So I messed up somewhere. Any help would be appreciated.

Answer 1

You didn't mess up, you forgot to notice another fact. The $x_i$ are not only orthogonal, but in fact orthonormal