Let $C(D) \cap H(\bar D)$ denote the inner product space of functions these are analytic in unit disk $D$ and continuous in $\bar D$, equipped with the inner product $(f,g)= \frac{1}{2 \pi} \int\_{0}^{2 \pi} f(e^{i \theta}) \overline {g(e^{i \theta})} d \theta$, is it a Hilbert space? 



A problem on completeness of a specific space of complex functions
No, it is not complete. Completion of this space is called $H^2(D)$.
It consists of all analytic functions for which
$$\| f\|^2:=\sup\_r\frac{1}{2\pi}\int\_0^{2\pi}|f(re^{i\theta})|^2d\theta$$
is finite, or alternatively
$$f(z)=\sum\_0^\infty a\_nz^n,\quad \sum\_0^\infty|a\_n|^2<\infty,$$
or alternatively, all those function in $L\_2(-\pi,\pi)$ whose negative Fourier
coefficients vanish.


For an example of a function which is in $H^2$ but not continuous in the closed disk, take $$\exp\left(\frac{z+1}{z-1}\right)$$
which is bounded in the unit disk, therefore belongs to $H^2$ but is not
continuous because the boundary values have absolute value $1$ except at $z=1$
where the radial limit is $0$. 


