A question on strictly cosingular operators Let $T:X\rightarrow Y$ be an operator satisfying that $Q_{N}T$ is not surjective for every finite-dimensional subspace $N$ of $Y$, where $Q_{N}:Y\rightarrow Y/N$ is the canonical quotient map. My question is: given any $\epsilon>0$, is there an infinite-codimensional subspace $M$ with a Schauder basis such that $Q_{M}T$ is compact and the norm of $Q_{M}T$ is less than $\epsilon$?
I have another elementary question: Let $M$ be a closed subspace of $X^{*}$. Is the closed unit ball of $\overline{M}^{w*}$ necessarily contained in $\overline{B_{M}}^{w*}$?
Thank you!
 A: Your first question is silly as stated. Consider $0\oplus I$ on $\ell_2 \oplus \ell_\infty$. 
As for the second question, Google "total non norming subspaces" to find a wealth of counterexamples.
Added 11/29/16: Oh, so Pietsch does not claim that the subspace $M$ is separable.  For your first question to make sense you should assume that $Y$ is separable.  Even so, the answer is negative: Consider $0\oplus I$ on $X:= \ell_2\oplus Z$, where $Z$ is a separable space that fails the approximation property (AP).  Suppose that $M$ is a subspace of $X$ that has the AP and $\|{Q_M}_{|Z}\| < \epsilon$ for some small enough $\epsilon$ to make the claim below true.  Let $P_Z$ be the natural projection from $X$ onto $Z$. 
Claim: $P_Z M = Z$. 
Assume the claim. Let $W:= \ell_2 \cap M$.  Since $W$ is necessarily complemented in $X$, you can write $M$ as the direct sum of $W$ and some closed subspace $M_0$ of $M$, and clearly $P_Z$ is injective on $M_0$ and $P_Z M_0 = Z$.  Thus $M_0$ is isomorphic to $Z$ and hence $M_0$, whence also $M$, must fail the approximation property.
Proof of claim. There should be a book reference for this, but I don’t know one. The argument I give is basically the proof of what is often called the ``little open mapping theorem”. By the hypothesis that $\|{Q_M}_{|Z}\| < \epsilon$, for each non zero $z\in Z$ there is $x_z \in M$ s.t. $\| z- x_z\| < \epsilon \|z\|$, and necessarily $\|x_z\| < (1+\epsilon)\|z\|$. Since $\|P_Z\| = 1$, we also have $\|z - P_z x_z\| < \epsilon$. Repeat this, replacing $z$ by $z - P_Z x_z \in Z$ and iterate in the obvious way.  This successive approximation argument produces a sequence $(x_n)$ in $M$ s.t. $\|x_n\| < \epsilon^{n-1} (1+\epsilon) \|z\|$ and $z = \sum P_Z x_n$. So you just need $\epsilon < 1$.
A: On your second question: by now the answer is well known, but this question was first asked by S. Banach in terms of iterated weak* sequential closures and answered by S. Mazurkiewicz [Studia Math. 2 (1930), 68–71]. Later this result of Mazurkiewicz was developed in many different directions, mostly using the term  "total non-norming subspace" mentioned by Bill Johnson. You can find some account on these developments (as of 2000) in my survey https://arxiv.org/abs/math/0203139.
