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Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, the series $\sum\limits_{n=1}^{\infty}a_{n}x_{n}$ converges in norm.

Let $(x_{n})_{n}$ be a bounded sequence in a Banach space $X$. We set

$$ \textrm{ca}((x_{n})_{n})=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$

Then $(x_{n})_{n}$ is norm-Cauchy if and only if $\textrm{ca}((x_{n})_{n})=0$.

Let $(x_{n})_{n}$ be a basis for a Banach space $X$. We introduce a quantity measuring non-bounded completeness as follows: $$\textrm{bc}((x_{n})_{n})=\sup\Big\{\textrm{ca}((\sum_{i=1}^{n}a_{i}x_{i})_{n})\colon (\sum_{i=1}^{n}a_{i}x_{i})_{n}\subseteq B_{X}\Big\},$$ where $B_{X}$ is the closed unit ball of $X$. Clearly, $(x_{n})_{n}$ is boundedly complete if and only if $\textrm{bc}((x_{n})_{n})=0$.

We think about the $\textrm{bc}$-values of some known bases and obtain the following:

1. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$.
2. $\operatorname{bc}((s_{n})_{n})=1$, where $(s_{n})_{n}$ is the summing basis of $c_{0}$.
3. $\operatorname{bc}((e_{n})_{n=0}^{\infty})=2$, where $(e_{n})_{n=0}^{\infty}$ is the unit vector basis of $c$ ($e_{0}=(1,1,1,\ldots)$).
4. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of the James space $\mathcal{J}$.
5. Let $x_{1}=e_{1}$ and $x_{n}=-x_{n-1}+(n-1)e_{n}$ for $n\geq 2$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$. Then $\operatorname{bc}((x_{n})_{n})=1$.
6. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Haar basis of $L_{1}[0,1]$.
7. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Faber-Schauder basis of $C[0,1]$.

The examples above yield naturally the following question:

Question. $\textrm{bc}((x_{n})_{n})=1$ or $2$ for every basis $(x_{n})_{n}$ that is not boundedly complete ?

Prof. William B. Johnson has the following guess with respect to Question:

Guess. Let $(x_{n})_{n}$ be a basis for a Banach space $X$ that is not boundedly complete. If $(x_{n})_{n}$ is monotone and shrinking, then $\textrm{bc}((x_{n})_{n})=1$ or $2$.

I can not prove Guess. Thank you !

Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, the series $\sum\limits_{n=1}^{\infty}a_{n}x_{n}$ converges in norm.

Let $(x_{n})_{n}$ be a bounded sequence in a Banach space $X$. We set

$$ \textrm{ca}((x_{n})_{n})=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$

Then $(x_{n})_{n}$ is norm-Cauchy if and only if $\textrm{ca}((x_{n})_{n})=0$.

Let $(x_{n})_{n}$ be a basis for a Banach space $X$. We introduce a quantity measuring non-bounded completeness as follows: $$\textrm{bc}((x_{n})_{n})=\sup\Bigg\{\textrm{ca}\Big(\big(\sum_{i=1}^{n}a_{i}x_{i}\big)_{n}\Big)\colon \big(\sum_{i=1}^{n}a_{i}x_{i}\big)_{n}\subseteq B_{X}\Bigg\},$$ where $B_{X}$ is the closed unit ball of $X$. Clearly, $(x_{n})_{n}$ is boundedly complete if and only if $\textrm{bc}((x_{n})_{n})=0$.

We think about the $\textrm{bc}$-values of some known bases and obtain the following:

1. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$.
2. $\operatorname{bc}((s_{n})_{n})=1$, where $(s_{n})_{n}$ is the summing basis of $c_{0}$.
3. $\operatorname{bc}((e_{n})_{n=0}^{\infty})=2$, where $(e_{n})_{n=0}^{\infty}$ is the unit vector basis of $c$ ($e_{0}=(1,1,1,\ldots)$).
4. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of the James space $\mathcal{J}$.
5. Let $x_{1}=e_{1}$ and $x_{n}=-x_{n-1}+(n-1)e_{n}$ for $n\geq 2$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$. Then $\operatorname{bc}((x_{n})_{n})=1$.
6. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Haar basis of $L_{1}[0,1]$.
7. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Faber-Schauder basis of $C[0,1]$.

The examples above yield naturally the following question:

Question. $\textrm{bc}((x_{n})_{n})=1$ or $2$ for every basis $(x_{n})_{n}$ that is not boundedly complete ?

Prof. William B. Johnson has the following guess with respect to Question:

Guess. Let $(x_{n})_{n}$ be a basis for a Banach space $X$ that is not boundedly complete. If $(x_{n})_{n}$ is monotone and shrinking, then $\textrm{bc}((x_{n})_{n})=1$ or $2$.

I can not prove Guess. Thank you !

Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, the series $\sum\limits_{n=1}^{\infty}a_{n}x_{n}$ converges in norm.

Let $(x_{n})_{n}$ be a bounded sequence in a Banach space $X$. We set

$$ \textrm{ca}((x_{n})_{n})=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$

Then $(x_{n})_{n}$ is norm-Cauchy if and only if $\textrm{ca}((x_{n})_{n})=0$.

Let $(x_{n})_{n}$ be a basis for a Banach space $X$. We introduce a quantity measuring non-bounded completeness as follows: $$\textrm{bc}((x_{n})_{n})=\sup\Big\{\textrm{ca}((\sum_{i=1}^{n}a_{i}x_{i})_{n})\colon (\sum_{i=1}^{n}a_{i}x_{i})_{n}\subseteq B_{X}\Big\},$$ where $B_{X}$ is the closed unit ball of $X$. Clearly, $(x_{n})_{n}$ is boundedly complete if and only if $\textrm{bc}((x_{n})_{n})=0$.

We think about the $\textrm{bc}$-value of some known bases and obtain the following:

1. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$.
2. $\operatorname{bc}((s_{n})_{n})=1$, where $(s_{n})_{n}$ is the summing basis of $c_{0}$.
3. $\operatorname{bc}((e_{n})_{n=0}^{\infty})=2$, where $(e_{n})_{n=0}^{\infty}$ is the unit vector basis of $c$ ($e_{0}=(1,1,1,\ldots)$).
4. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of the James space $\mathcal{J}$.
5. Let $x_{1}=e_{1}$ and $x_{n}=-x_{n-1}+(n-1)e_{n}$ for $n\geq 2$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$. Then $\operatorname{bc}((x_{n})_{n})=1$.
6. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Haar basis of $L_{1}[0,1]$.
7. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Faber-Schauder basis of $C[0,1]$.

The examples above yields naturally the following question:

Question. $\textrm{bc}((x_{n})_{n})=1$ or $2$ for every basis $(x_{n})_{n}$ that is not boundedly complete ?

Prof. William B. Johnson has the following guess with respect to Question:

Guess. Let $(x_{n})_{n}$ be a basis for a Banach space $X$ that is not boundedly complete. If $(x_{n})_{n}$ is monotone and shrinking, then $\textrm{bc}((x_{n})_{n})=1$ or $2$.

I can not prove Guess. Thank you !

Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, the series $\sum\limits_{n=1}^{\infty}a_{n}x_{n}$ converges in norm.

Let $(x_{n})_{n}$ be a bounded sequence in a Banach space $X$. We set

$$ \textrm{ca}((x_{n})_{n})=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$

Then $(x_{n})_{n}$ is norm-Cauchy if and only if $\textrm{ca}((x_{n})_{n})=0$.

Let $(x_{n})_{n}$ be a basis for a Banach space $X$. We introduce a quantity measuring non-bounded completeness as follows: $$\textrm{bc}((x_{n})_{n})=\sup\Big\{\textrm{ca}((\sum_{i=1}^{n}a_{i}x_{i})_{n})\colon (\sum_{i=1}^{n}a_{i}x_{i})_{n}\subseteq B_{X}\Big\},$$ where $B_{X}$ is the closed unit ball of $X$. Clearly, $(x_{n})_{n}$ is boundedly complete if and only if $\textrm{bc}((x_{n})_{n})=0$.

We think about the $\textrm{bc}$-values of some known bases and obtain the following:

1. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$.
2. $\operatorname{bc}((s_{n})_{n})=1$, where $(s_{n})_{n}$ is the summing basis of $c_{0}$.
3. $\operatorname{bc}((e_{n})_{n=0}^{\infty})=2$, where $(e_{n})_{n=0}^{\infty}$ is the unit vector basis of $c$ ($e_{0}=(1,1,1,\ldots)$).
4. $\operatorname{bc}((e_{n})_{n})=1$, where $(e_{n})_{n}$ is the unit vector basis of the James space $\mathcal{J}$.
5. Let $x_{1}=e_{1}$ and $x_{n}=-x_{n-1}+(n-1)e_{n}$ for $n\geq 2$, where $(e_{n})_{n}$ is the unit vector basis of $c_{0}$. Then $\operatorname{bc}((x_{n})_{n})=1$.
6. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Haar basis of $L_{1}[0,1]$.
7. $\operatorname{bc}((f_{n})_{n})=2$, where $(f_{n})_{n}$ is the Faber-Schauder basis of $C[0,1]$.

The examples above yield naturally the following question:

Question. $\textrm{bc}((x_{n})_{n})=1$ or $2$ for every basis $(x_{n})_{n}$ that is not boundedly complete ?

Prof. William B. Johnson has the following guess with respect to Question:

Guess. Let $(x_{n})_{n}$ be a basis for a Banach space $X$ that is not boundedly complete. If $(x_{n})_{n}$ is monotone and shrinking, then $\textrm{bc}((x_{n})_{n})=1$ or $2$.

I can not prove Guess. Thank you !