Let $(V, \|\cdot\|)$ be a Banach space over $\mathbf R$ and $W$ be a closed subspace of $V$. We know that $V/W$ becomes a normed linear space under the quotient norm $\|\cdot\|_q$ defined as $\|v+W\|_q=\inf_{w\in W} \|v+w\|$ for all $v\in V$.
Suppose $v\in V$ and we have $\|{v+W}\|_q=\lambda$. Does there necessarily exist a $w\in W$ such that $\|{v+w}\|=\lambda$.
Clearly, the above is true if $\lambda=0$ or if $V$ is finite dimensional (because we can use the compactness of $S^n$).
But I am getting nowhere with the statement in block quotes.
We get a sequence $(w_n)$ in $W$ such that $\|v+w_n\|>\lambda+1/n$ for all $n$. But this is weaker than saying $\|v+W\|_q=\lambda$.
