A question on the norm of the restriction of an operator to subspaces 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: 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.
A: When $N=\overline{M}$ clearly we have $$||T_{N}||=||T|_{\overline{M}}||\geqslant||T_{M}||\geqslant ||T|_W||$$
for any closed subspace $W\subseteq M$. Also analogue when $\overline{M}=M$. 
When $N\subset \overline{M}$ is a proper subspace then since $N=\overline{N}$ (closed) then it must be the case $N\subseteq M$ since $\overline{M}$ is the smallest closed set containing $M$. Let $W\subseteq M$ be some closed subspace. If $N\cap W=\emptyset$ then either case is possible $||T_N||\geqslant ||T_W||$ or $||T_W||\geqslant ||T_N||$. If $N\cap W\neq\emptyset$ then $N\cap W\subseteq M$ is closed subspace (both $N$ and $W$ are closed subspaces). Then $$||T_{N\cap W}||\leqslant\min\{||T_N||,||T_W||\}\leqslant ||T_N||$$
So $N\cap W$ fullfills the condition whenever it is nonempty and infinite dimensional. 
