Let $X$ be a Banach space and $M$ a closed linear subspace of $X$. Define the equivalence relation $x \sim y$ if $x - y \in M$. The resulting quotient space, $X/M$, is another Banach space.
If $X$ were a Hilbert space, we could decompose $X$ as $X = M \oplus M^\perp$. In their book, Reed & Simon (volume I) say that the Banach space $X/M$ can "sometimes take the place of $M^\perp$ in the Banach case where there is no orthogonality".
I thought this was an interesting comment, but I did not fully understand what was meant by it. How does the quotient Banach space $X/M$ compare to $M^\perp$?