习题 14, § 5

令 $E$ 为 Banach 空间满足可数性第一公理。 如果满足以下性质，$E$的元素的序列$(e_n)_{n \geq 0}$被称为*Banach基*：对于每个$x \in E$，存在一个唯一的标量序列$(a_n)$，使得$x = \sum_{n=0}^\infty a_n e_n$，其中右侧的级数是收敛的。 a) 表明家庭$(e_n)$是完整且自由的。 令$E_n$为由$e_m$为指标$m \leq n$生成的$E$的向量子空间（闭合），并令$P_n$为从$E$到由$P_n \cdot (\sum_{m=0}^\infty a_m e_m) = \sum_{m=0}^n a_m e_m$定义的$E_n$的投影算子。 Show that the $P_n$ are continuous linear mappings and that $\sup_n \|P_n\| < +\infty$. (Consider the norm on $E$ defined by $\left\| \sum_{n=0}^\infty a_n e_n \right\| = \sup_n \left\| \sum_{m=0}^n a_m e_m \right\|$; show that $E$ is complete for this norm and deduce that it is equivalent to the given norm on $E$ (cf. I, p. 17, th. 1).) Show that for every pair of integers $p < q$, the norm of the projection $P_{q,p}$ of $E_q$ onto $E_p$, which is parallel to the vector subspace generated by the $e_m$ for indices $m$ such that $p + 1 \leq m \leq q$, is bounded by a number independent of $p, q$. b) Conversely, let $(e_n)$ be a total sequence of elements of $E$, which is a free family and is such that the norms of the projections $P_{q,p}$ for $0 \leq p < q$ are bounded by a number $M$ independent of $p$ and $q$. 证明 $(e_n)$ 是 $E$ 的 Banach 基。 （首先证明序列$(e_n)$是拓扑自由的并且定义投影算子$P_n$；然后$\|P_n\| \leq M$对于所有$n$。 接下来观察，如果$d(x, E_n)$是点$x \in E$到$E_n$的距离，则$\|x - P_n \cdot x\| \leq (M + 1)\ d(x, E_n)$。）[^1]

[^1]: It is clear that the existence of a Banach basis in $E$ implies that $E$ satisfies the first axiom of countability. 但是 Banach 空间满足的例可数性第一公理，其中不存在 Banach 基 (P. Enflo, Acta Math., t. CXXX（1973），p. 309-317）。