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?


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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.