Let $X,Y$ be infinite-dimensional Banach spaces and $T:X \rightarrow Y$ be a bounded linear operator. Let $M$ be an infinite-dimensional subspace of $X$ ($M$ is not necessarily closed). Let $N$ be an infinite-dimensional closed subspace of $\overline{M}$. My question is the following:


Question. Is there an infinite-dimensional closed subspace $W$ of $M$ such that $\|T|\_{W}\|\leq \|T|\_{N}\|$? 



A question on the norm of the restriction of an operator to subspaces
In general, no. 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 both infinite dimension and infinite co-dimension. Then, there is an infinite dimensional dense subspace $M\subset X$ of $N$ such that $M\cap N=(0)$ and $N\subset \overline{M} $. In this situation, of course, any *non-null* subspace $W\subset M$ verifies $\|T\_{|W}\| >0=\|T\_{|N}\|$. 


**Details.** Finding the subspace $M$ requires a little argument. By the assumptions on $N$: There is a sequence $\{u\_k\}\_k$ such that $\overline{\operatorname{span}}\{u\_k\}\_k=N$. There is an infinite dimensional subspace $N'$ such that $N'\cap N=(0)$. There is a bounded linearly independent double sequence $\{v\_{j,k}\}\_{j,k}\subset N'$. Then $\{u\_k+2^{-j}v\_{j,k}\}\_{j,k}$ is a linearly independent family that generates a linear subspace $M$ such that $N\subset \overline{M}$ and $N\cap M=(0)$, as wanted.


