We introduce a new class $$\mathcal {FV}(\Omega ,E)$$ of weighted spaces of functions on a set $$\Omega$$ with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smooth, holomorphic or harmonic functions. Then we exploit the construction of $$\mathcal {FV}(\Omega ,E)$$ to derive sufficient conditions such that $$\mathcal {FV}(\Omega ,E)$$ can be linearised, i.e. that $$\mathcal {FV}(\Omega ,E)$$ is topologically isomorphic to the $$\varepsilon$$-product $$\mathcal {FV}(\Omega )\varepsilon E$$ where $$\mathcal {FV}(\Omega ):=\mathcal {FV}(\Omega ,\mathbb {K})$$ and $$\mathbb {K}$$ is the scalar field of E.

中文翻译：

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向量值函数的加权空间和 $$\varepsilon$$-product

我们在集合 $$\Omega$$ 上引入了一个新类 $$\mathcal {FV}(\Omega ,E)$$ 函数的加权空间，其值位于局部凸 Hausdorff 空间 E 中，该空间覆盖了许多经典的向量空间-值函数，如连续、平滑、全纯或调和函数。然后我们利用 $$\mathcal {FV}(\Omega ,E)$$ 的构造推导出充分条件使得 $$\mathcal {FV}(\Omega ,E)$$ 可以线性化，即 $$ \mathcal {FV}(\Omega ,E)$$ 与 $$\varepsilon$$-积 $$\mathcal {FV}(\Omega )\varepsilon E$$ 拓扑同构，其中 $$\mathcal {FV} (\Omega ):=\mathcal {FV}(\Omega ,\mathbb {K})$$ 和 $$\mathbb {K}$$ 是 E 的标量场。