I'm reading a book on PDE in which the min-max theorem and the Rayleigh quotient are mentioned. In the 'compact operators' section in the given link, it seems to me that the proof employs the following proposition.
Proposition Let $H$ be an infinite dimensional Hilbert space with a countable , linearly independent subset $B=\{u_i:i=1,2,\dots\}$. Let $k$ be any positive integer.
If $B$ is in fact a basis for $H$, it is true that
(a) Let $S_k$ be any $k$-dimensional subspace of $H$ and let $S'=\overline{span\{u_i:i\geq k\}}$. Then $S'\cap S_k\neq \{0\}$.
(b) Let $V_{k-1}$ be any $k-1$-dimensional subspace of $H$ and let $V'=span\{u_i:1\leq i\leq k\}$. Then $V'\cap {V_{k-1}}^{\perp}\neq \{0\}$.
I know that this probably is very trivial but I just cannot see why they hold since I haven't touched linear algebra for a long time. If $H$ is finite-dimensional, say $\dim(H)=n$, then the proposition is clearly true because we know that $$\dim(S'\cap S_k)=\dim(S')+\dim(S_k)-\dim(S'+S_k)\geq (n-k+1)+k-n=1>0$$ $$\dim(V'\cap {V_{k-1}}^{\perp})\geq k+[n-(k-1)]-n=1>0$$
But how do we perform dimension counting arguments when $H$ is infinite-dimensional? Thank you.