Counting balls in Hilbert spaces

Let $W$ be a real Hilbert space of dimension $n$ and $V$ a Hilbert subspace of dimension $m$. Assume that $f_1,\cdots,f_k$ are points in $W$ such that the following holds:

1. there exists $\sigma>0$ such that $dist(f_i,V)\geq\sigma$ for $i=1,2,\cdots,k$,
2. there exists $0<\delta\leq\sigma$ such that the balls $B(f_i,\delta)$ are disjoint for $i=1,2,\cdots,k$,
3. there exists $L>0$ such that $||f_i||_2\leq L$ for $i=1,2,\cdots,k$.

Can you find a good bound on $k$ (the number of the points $f_i$) involving $n,m,\sigma,\delta,L$?

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