Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$ then by Uniform Boundedness principle we can observe that $$\sup_n||S_n||< \infty. $$

And the number $K=\sup_n ||S_n||$ is called basis constant.

At this point, I have a question. What is the importance of the basis constant $K$? What happens when $K=1$?