Let $M$ be a compact differentiable manifold and $\Delta : C^{\infty}(M) \to C^{\infty}(M)$. I am reading a lemma that states in some point of the proof:
On a compact manifold, the Laplacian has finite kernel (it is easy from Hodge theory). Even more, all non-zero eigenvalues of $\Delta$ are positive and form a simple specter, so we can pick the smallest one.
What does mean this sentence? What does mean a simple specter? Why can we pick the smallest eigenvalue?
The spectrum of a linear operator is said to be simple when there are no degenerate eigenspaces; that is, more than one eigenvector associated to a given eigenvalue.