Why in a finite dimensional space every orthonormal basis is basis

Question

Why in a finite dimensional space every orthonormal basis is basis

i know in infinite dimensional space every basis is orthonormal basis but converse is not true ( for example $l^2$ ) but in finite dimensional converse is true why ?

Answer 1

I will assume that what you are talking about here are Hilbert bases.

In general, in a Hilbert space $H$, you can indeed have an orthonormal set $\{e_\lambda\mid\lambda\in\Lambda\}$ which is a Hilbert basis of $H$, but which is not a Hamel basis (that is, a basis in the Linear Algebra sense). Such a set is always linearly independent.

But if $\dim H<\infty$, then the set $\{e_\lambda\mid\lambda\in\Lambda\}$ is finite too. And then, asserting that each $v\in H$ can be written as $\sum_{\lambda\in\Lambda}a_\lambda e_\lambda$ just means that every $v\in H$ is a linear combination of elements of $\{e_\lambda\mid\lambda\in\Lambda\}$. And therefore this set actually spans $H$.