Abstract For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are not identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces and . In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to -Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where and are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of -Banach frames make sense.

中文翻译：

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希尔伯特空间中具有固定对偶的算子方程解的框架

摘要 对于算子方程的求解，Stevenson 引入了框架的定义，其中不识别 Hilbert 空间及其对偶。这意味着 Riesz 同构不用作标识，例如，这对 Sobolev 空间和 没有意义。在本文中，我们将重新审视 Stevenson 框架的概念并将其引入 Banach 空间。这相当于 -Banach 帧。已知，如果存在这样的系统，通过定义新的内积并使用 Riesz 同构，Banach 空间与 Hilbert 空间同构。在本文中，我们处理对比设置，其中 和 未识别，并区分等效规范，并表明在这种设置中对 -Banach 框架的研究是有意义的。