If $V$ is a finite dimensional inner product space of dimension $n$ and $e_k$ an orthonormal basis, for $x\in V$, why is $x=\sum_{k=1}^{n}\langle x,e_k\rangle e_k$?
In my lecture notes the professor used this fact as part of a proof of a theorem and I can't see why it's true. It's probably trivial but I just can't prove it. All I've done is written $x$ as
$$x=\langle x,e_1\rangle e_1 +…+\langle x,e_n\rangle e_n$$ but it's still not obvious to me.
Best Answer
You can write $x$ as $\sum_{k=1}^na_ke_k$. But then, since $\langle e_j,e_j\rangle=1$ and $\langle e_j,e_k\rangle=0$ if $k\ne j$,$$a_j=\left\langle\sum_{k=1}^na_ke_k,e_j\right\rangle=\langle x,e_j\rangle.$$