# “Orthogonal complement” of a subspace of a Banach space

I'm looking for a reference. I am considering the situation where $X$ is a Banach space and $Y$ is a closed finite co-dimensional subspace. I am looking for a $W$ that is a complement of $Y$ (i.e. so that $X$ is the topological direct sum of $W$ and $Y$). I want them to be as far from parallel as possible.

The sense that I'm toying with (but I'm happy to have other senses proposed - I suspect they're all fairly equivalent) is that I want to consider the map $\Phi\colon Y\times W\to X$ given by $\Phi(y,w)=y+w$ to have the property that $\|\Phi^{-1}\|$ has small norm (say you put the norm $\|(y,w)\|=\max(\|y\|,\|w\|)$ on the product.

So I'm looking for $W$ that makes $\|\Phi^{-1}\|$ small. The question then is: how small can you make this quantity? I think I can make it less than $6^n$ if $Y$ is $n$-codimensional (irrespective of the space $X$), but I presume there are better bounds and references out there.

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You can make the norm of $\|\Phi^{-1}\|$ to be of order $\sqrt n$. This is basically a theorem of Kadets and Snobar. A good reference is III.B.11 in Wojtaszczyk, P., Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25. Cambridge University Press, Cambridge, 1991.