Rephrasing statements in terms of Eigenfunctions, Eigenvalues and Eigenspaces of Differential Operator

Question

Let $V$ be a set of functions that are infinitely differentiable, real, $2\pi$-periodic functions. Also, let $D = \frac{d^2}{dx^2}$ be the second derivative function that is a linear transformation.

What does it mean for the following in terms of eigenvalues, eigenfunctions, and eigenspaces of D:

• Let $\lambda \in \mathbb{R}$. If $f \in V$ satisfies $\forall x \in \mathbb{R}, \, f''(x) = \lambda f(x)$ then $\lambda = \lambda_n$ for some $n \in \mathbb{N}^+$.

• Let $n \in \mathbb{N}^+$. If $f \in V$ satisfies $\forall x \in \mathbb{R}, \, f''(x) = \lambda_n f(x)$ then there exists unique $A,B \in \mathbb{R}$ such that $\forall x \in \mathbb{R}, \, f(x) = A \sin(n x) + B \cos(n x)$.

• If $f \in V$ satisfies $\forall x \in \mathbb{R}, f''(x) = 0$ then $f$ is constant.

I need a conceptual breakdown of what these really mean. I'm having trouble wrapping my head around it.

Answer 1

It is easy to check that $D$ is a linear map from $V$ to $V$. The first statement claims that the set of eigenvalues of $D$ is countable. The second means for each eigenvalue $\lambda_n=-n^2>0$ the linear space of the eigenvectors with eigenvalues is two-dimensional with a basis $\{\sin nx,\cos nx\}$. The third statement means that the linear space of the eigenvectors with eigenvalue $0$ is an one-dimensional space of constant functions.