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Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. It is important to know the exact expression of the norm of $\|\sum_{i=1}^{n}a_{i}x_{i}\|$ for all $n$ and all scalars $a_{1},a_{2},\ldots,a_{n}$. For example, if $(s_{n})_{n}$ is the summing basis of $c_{0}$, then $$\Big\|\sum_{i=1}^{n}a_{i}s_{i}\Big\|=\max_{1\leq k\leq n}\Big|\sum_{i=k}^{n}a_{i}\Big|;$$ If $(e_{n})_{n=0}^{\infty}$ is the unit vector basis of $c$ ($e_{0}=(1,1,1,\ldots)$), then $$\Big\|\sum_{i=0}^{n}a_{i}e_{i}\Big\|=\max\big(|a_{0}|,|a_{0}+a_{1}|,\ldots,|a_{0}+a_{n}|\big).$$ These expressions are quite useful. My concern is the expression of $\|\sum_{i=1}^{n}a_{i}h_{i}\|$, where $(h_{n})_{n}$ is the Haar basis for $L_{1}[0,1]$.

Question 1. $\|\sum_{i=1}^{n}a_{i}h_{i}\|=$ ?

Question 2. $\|\sum_{i=0}^{n}a_{i}f_{i}\|=$ ?, where $(f_{n})_{n=0}^{\infty}$ is the Faber-Schauder basis for $C[0,1]$.

Thank you !

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  • $\begingroup$ What do you get for $\|\sum_{i=1}^{2}a_{i}h_{i}\|$ and $\|\sum_{i=1}^{3}a_{i}h_{i}\|$ ? $\endgroup$
    – Gerald Edgar
    Commented Jan 14, 2022 at 10:46
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    $\begingroup$ Haar basis is usually indexed by two indices as $h_{j,k}$, $j\in\mathbb{N}$ and $k\in\mathbb{Z}$. It might be helpful to use double indices. $\endgroup$
    – Onur Oktay
    Commented Jan 14, 2022 at 16:34
  • $\begingroup$ @OnurOktay You are right. I shall use double indices. $\endgroup$
    – Dongyang Chen
    Commented Jan 15, 2022 at 3:48
  • $\begingroup$ @GeraldEdgar I shall first consider $\|\sum_{i=1}^{2}a_{i}h_{i}\|$ and $\|\sum_{i=1}^{3}a_{i}h_{i}\|$, and then by induction, maybe... $\endgroup$
    – Dongyang Chen
    Commented Jan 15, 2022 at 3:51

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