Question

Hello, I'd like some help to find an answer I've been looking for since this morning. Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ onto $Y$. Which is the best estimate for the norm of $P$? I found this information in an article by Bohnenblust as far as $n=1$ is concerned (that is, there always exists a projection $P$ such that $\|P\|\leq 2+\varepsilon$), but nothing satisfactory when the codimension increases. Thank you.

Answer 1

In many books$^*$ you can find the result that there is a projection of norm at most $\sqrt{n}$ onto any $n$ dimensional subspace of a Banach space. For reflexive spaces, this gives immediately that every $n$ codimensional subspace is the range of a projection that has norm at most $\sqrt{n} +1$. For non reflexive spaces, by using the principle of local reflexivity (which also is in many books), you get for any $\epsilon > 0$ the estimate $\sqrt{n} +1 + \epsilon $. $$ $$ $*$ See, for example, Albiac and Kalton, ``Topics in Banach space theory", Theorem 12.1.6. In this book you can also find the principle of local reflexivity.

Answer 2

The obvious answer is: $2^n +\varepsilon$. Just iterate the Bohnenblust construction to $X\supset Y_{n-1}\supset \dots \supset Y_1 \supset Y_0=Y$ where $y_1,\dots ,y_n \in X$ span a complement of $Y$ and $Y_i$ is the span of $Y$ and $y_1,\dots,y_i$.