Consider an operator $A: H\rightarrow H,$ where $H\equiv L_2[0; \pi],$ such that $D(A)=C_0^2[0;\pi],$ and is defined by
\begin{eqnarray}\label{eqn3-0}
\left\{
\begin{array}{lll}
\displaystyle A(U(x))\equiv\frac{d^2U(x)}{dx^2},\\
U(0)=0,\quad U(\pi)=0.
\end{array}
\right.
\end{eqnarray}
Let us construct the closure of operator $A$ in the space $L_2[0;\pi]$ by introducing the scalar product.
\begin{definition}
The scalar product in space $L_2[0;\pi]$ is integral
\begin{eqnarray*}
(W(x), U(x))=\int\limits_0^{\pi}U(x)W(x)dx,
\end{eqnarray*}
where $W(x), U(x)\in L_2[0;\pi].$
\end{definition}
Operator $A$ is symmetric on its domain. The symmetry of the operator implies its closure in $L_2[0;\pi].$ Since the operator is symmetric on $D(A),$ then, according to \cite{Lusternik}, p. 354
\begin{definition}
The closure of Operator $A$ is Operator $\bar{A}$ such that $D(A)\subset D(\bar{A}),$ and the graph of $\Gamma(\bar{A})$ is closed in the Cartesian product $L_2[0; \pi]\times L_2[0; \pi].$
\end{definition}
The solution to (\ref{eqn1-1}) has the form of a series in eigenfunctions of the Sturm-Liouville problem
\begin{eqnarray}\label{eqn3-1}
\left\{
\begin{array}{lll}
\displaystyle \frac{d^2U(x)}{dx^2}=\Lambda U(x), x\in (0; \pi),\\
\displaystyle U(0)=0, U(\pi)=0.
\end{array}
\right.
\end{eqnarray}
\begin{lemma}\label{lemma1}
The eigenvalues of the Sturm-Liouville problem (\ref{eqn3-1}) are $\Lambda=-n^2.$ The system of eigenfunctions corresponding to these numbers $\left\{\sin( nx)\right\}_{n=1}^{\infty}$ is complete and orthogonal in the space $L_2[0; \pi].$
\end{lemma}
\begin{proof}[Proof:]
The proof is given in, for example, \cite{Steklov}.
\end{proof}