I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem
$$\left\{ \begin{aligned} -u''(x)&=\lambda u(x)\quad 0<x<\pi\\ u(0)&=0\\ u(\pi)&=0. \end{aligned} \right.$$
I think I might use the fact that the set of eigenfunctions $\{\sin k x\}_{k=1}^{+\infty}$ is complete in $L^2((0,\pi))$
Lemma. (Minimax Principle, Version 2) $$\lambda_n = \inf_{X \in \Phi_n(H_0^1)} \{ \sup_{u \in X} \rho(u) \}$$ where $\Phi_n(H_0^1)=\{n-\text{dimensional linear subspace } X \subset H_0^1 \}$.
Precision : $\rho(u)$ is the Rayleigh quotient
Clearly, I know that the $1$ and $2$-dimensional linear subspaces $X \subset \mathbb{R}^3$ are the lines and planes in $\mathbb{R}^3,$ but how could we interpret, for instance, a $2$-dimensional linear subspace $X \in \Phi_n(H_0^1)$ (Complete answer would be appreciated)?
Thanks for your help!