We extend the algebraic construction of finite dimensional varying exponent \(L^{p(\cdot )}\) space norms, defined in terms of Cauchy polynomials to a more general setting, including varying exponent \(L^{p(\cdot )}\) spaces. This boils down to reformulating the Musielak–Orlicz or Nakano space norm in an algebraic fashion where the infimum appearing in the definition of the norm should become a (uniquely attained) minimum. The latter may easily fail, as turns out, and in this connection we examine the Fatou type semicontinuity conditions on the modulars. Norms defined by ODEs are applied in studying such semicontinuity properties of \(L^{p(\cdot )}\) space norms with \(p(\cdot )\) unbounded.

我们将用柯西多项式定义的有限维变化指数\（L ^ {p（\ cdot）} \）空间范数的代数构造扩展到更一般的设置，包括变化指数\（L ^ {p（\ cdot ）} \）空格。这归结为以代数方式重新构造Musielak-Orlicz或Nakano空间范数，其中范数定义中出现的最小值应成为（唯一达到的）最小值。事实证明，后者可能很容易失效，因此，我们在此检查了模块上的Fatou型半连续条件。由ODE定义的范数适用于研究\（p（\ cdot）\）无界的\（L ^ {p（\ cdot）} \）空间范数的半连续性。