simple harmonic oscillator functions space and real plane the functions that satisfy the simple harmonic oscillator equation $\frac{\mathrm{d}^2x }{\mathrm{d} t^2}=-\omega _{0}^2x$ constitute a linear space . is this linear space is isomorphic to $\mathbb{R}^2$ ?

This might be a bit advanced for you now, but eventually you may study linear systems of differential equations. They take the form $\mathbf{y}'(t) = A \mathbf{y}(t)$, where $\mathbf{y}(t)$ is a (column) vector of $n$ functions of $t$, and $A$ is an $n\times n$ matrix.

Your equation is one of these, because you can let $y_1(t) = x(t)$ and $y_2(t) = x'(t)$. Then $y_1' = y_2$ and $y_2' = -\omega_0^2 y_1$. That is, $$ \begin{bmatrix} y_1 \\\ y_2 \end{bmatrix}' =\begin{bmatrix} 0 & 1 \\\ -\omega_0^2 & 0 \end{bmatrix} \begin{bmatrix} y_1 \\\ y_2 \end{bmatrix} $$

The solution space of a linear system of $n$ differential equations has dimension $n$.