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Let $X$ be a Banach space and let $(e_n)_{n=1}^{\infty}$ be a Schauder basis for $X$. Let us denote the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$. Then by uniform boundedness principle one has $$\sup_n||S_n||<\infty.$$

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

At this point, I have a question. What is the importance of basis constant? What happens when $K=1$, i.e; what happens when the basis is monotone? Is there any condition on $K$ to observe an unconditional basis?

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I'm not sure this is what you want, but for what it's worth:

The basis constant can be characterized as the smallest $K$ so that for any choice of scalars $(a_n)$ and any integers $m<n$ we have $$ \biggl\Vert\sum_{i=1}^m a_i e_i\biggr\Vert\le K \biggl\Vert\sum_{i=1}^n a_i e_i\biggr\Vert. $$

The value of $K$ gives a measure of how "orthogonal" the basis is in the sense that it measures how drastically the basis elements can cancel with each other. (In fact, in Hilbert space, a basis has constant $K=1$ if and only if it is orthogonal in the Hilbert space sense).

The value of $K$ is also used in the hypothesis of various theorems, specifically "perturbation theorems". For example: if $(x_n)$ is a normalized basis of $X$ with basis constant $K$ and if $(y_n)$ is a sequence in $X$ such that $\sum\Vert x_n-y_n\Vert <1/(2K)$, then $(y_n)$ is a basis of $X$ equivalent to $(x_n)$.

One can also obtain a bound on the norm of the associated coefficient functions from the norms of the $e_i$ and $K$.


Basis with basis constant $K=1$ are called monotone and are special if only for the fact that not every Banach space with a basis has one.

Also, re the above comments, monotone bases were originally called "orthogonal bases". If $K=1$, the norm of $\sum_{i=1}^m a_i e_i$ cannot be decreased by adding a linear combination of basis elements $e_j$, $j>m$.


The value of $K$ will not in general indicate if a given basis is unconditional.

(In fact, given a Banach space $X$ with a basis, there is also a conditional basis of $X$. $X$ may be given an equivalent renorming so that this basis is monotone (as is the case with any basis).
Finding specific examples of a conditional monotone basis is not hard (e.g., in $\ell_1$: $x_1=e_1$, $x_n=e_n-e_{n-1}$, $n>1$).)


All the above can be found in Ivan Singer's Bases in Banach Spaces, Vol. 1.

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