Every set of orthogonal functions is complete?

Question

Suppose that the functions $ \phi_{1}, \ \phi_{2},...,\ \phi_{n},...\ \ $ are orthogonal on the interval $[a,b]$. That is

$$ \int_{a}^{b} \phi_{n}\phi_{m} dx = 0 \ \ \ \forall n\neq m$$

If $$ \lim_{N\rightarrow \infty}{ \int_{a}^{b} { \Big(f-\sum_{n=1}^{N}C_{n}\phi_{n} \Big)^2 } dx} = 0 $$

for every function $f$ with the property that ( $f$ is square-integrable. )$$ \int_{a}^{b} f^2 dx < \infty$$

we say that the set of functions $\{\phi_{1},\phi_{2,...} \}\ $ is complete. Where $C_{n}\ $ are the Fourier coefficients.

Now, I wonder if there exists are set of orthogonal functions that is not complete, or every set of orthogonal functions is complete?.

Answer 1

Take any set of complete orthogonal functions, and remove one of the elements. Then the resulting set will no longer be complete.

Edit: in particular, you cannot reach the element you removed (assuming it's nonzero a.e.).

Note:

$$\int_a^b(\phi_1-\sum_{n=2}^n C_n \phi_n)^2 dx=\int_a^b\phi_1^2+(\sum_{n=2}^NC_n \phi_n)^2 dx\ge \int_a^b\phi_1^2 dx>0$$

Answer 2

Let our universe be $L^2([0, 1]),$ i.e. $\langle f, g \rangle = \int_0^1 f(x) \, g(x) \, dx.$

Here are three examples of sets of orthogonal functions which do not form a complete set:

1. $\{\}$

2. $\{ x \mapsto 1 \}$

3. $\{ x \mapsto 1, x \mapsto x \}$