Suppose $V$ is a $d$-dimensional subspace of $\mathbb{R}^n$. By Kadec-Snobar theorem we know there exists a projection $P$ from $\mathbb{R}^n$ onto $V$ such that $\|P\|_1 \le \sqrt{d}$, where $\|\cdot\|_1$ is the operator $\ell_1$ norm $$ \sup_{x \neq 0} \frac{\|Px\|_1}{\|x\|_1}. $$
I am new to functional analysis. I am wondering if there is any simple construction with explicit geometric interpretation in this special case.
Thanks in advance.