Orthogonal complement of orthogonal complement

Solution 1:

Let

$$V:=\left\{ f:[0,1]\to\Bbb R\;;\; f\;\;\text{is continuous}\right\}\;\;\text{over}\;\;\Bbb R$$

and with the inner product

$$\langle f,g\rangle:=\int\limits_0^1f(x)g(x)dx$$

Let

$$U:=\{ f \in V\;;\;f(0)=0\}\implies U^\perp=\{0\}\;,\;\;U^{\perp\perp}=\ldots$$