how can we say that every complete orthogonal set in inner product space forms basis ?`

Question

I know that every orthogonal set in inner product space is linearly independent. so complete orthogonal set is linearly independent. In fact it is largest set which is "orthogonal". But how can we say that That orthogonal set with max size(complete) is largest linearly independent set??? There can be possibility that maximum size of orthogonal set = k(say) and max size of linearly independnet set is greater than k. I know from textbook that such case does not arise. But i am not able to get proof for this. Pls clarify!

Answer 1

This property is usually defined as follows:

We say that an orthonormal set $S \subseteq V$ is maximal if for every $x \in V$ we have:

$$\langle x, s\rangle = 0, \forall s \in S \implies x = 0$$

Notice that this implies the property that there does not exist an orthonormal strict superset of $S$.

Now let $\{e_1, \ldots, e_n\}$ be an orthonormal maximal set in $V$. We have:

$$\mathrm{span}\,\{e_1, \ldots, e_n\} \oplus \left(\mathrm{span}\,\{e_1, \ldots, e_n\}\right)^\perp = V$$

Take an arbitrary $e \in \left(\mathrm{span}\,\{e_1, \ldots, e_n\}\right)^\perp$. We have $\langle e, e_i\rangle = 0$, $\forall i \in \{1, \ldots, n\}$ so by maximality we have $e = 0$. Thus $\left(\mathrm{span}\,\{e_1, \ldots, e_n\}\right)^\perp = \{0\}$ so we have $V = \mathrm{span}\,\{e_1, \ldots, e_n\}$.

Therefore, $\{e_1, \ldots, e_n\}$ is an orthonormal basis for $V$.