We prove that \(L_2(\mathbb {R})\) contains a Schauder basis of non-negative functions. Similarly, for all \(1<p<\infty \), \(L_{p}(\mathbb {R})\) contains a Schauder basic sequence of non-negative functions such that \(L_p(\mathbb {R})\) embeds into the closed span of the sequence. We prove as well that if X is a separable Banach space with the bounded approximation property, then any set in X with dense span contains a quasi-basis (Schauder frame) for X.

中文翻译：

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$$ L_2 $$ L 2的肖德基，由非负函数组成

我们证明\（L_2（\ mathbb {R}）\）包含非负函数的Schauder基础。类似地，对于所有\（1 <p <\ infty \），\（L_ {p}（\ mathbb {R}）\）包含非负函数的Schauder基本序列，使得\（L_p（\ mathbb {R }）\）嵌入序列的封闭范围。我们证明，以及如果X是与有界逼近性可分离Banach空间，然后在任何一组X密集跨度包含准的基础上（Schauder不帧）为X。