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Let $T:X\to Y$ be a bounded linear operator between infinite dimensional Banach spaces, as assumed in the question. Assume further $N:=\ker T$ is a separable subspace of infinite dimension and co-dimension. Then, there is an infinite dimensional dense subspace $M\subset X$ such that $M\cap N=(0)$, and of course any non-null subspace $W\subset M$ verifies $\|T_{|W}\| >0=\|T_{|N}\|$. (And here $N$ is a closed infinite dimensional subspace of $\overline{M}=X$ as required).