Distribution Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space
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Theorem
Let $\map {\DD'} {\R^d}$ be the distribution space.
Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers.
Let $\paren +$ be the pointwise addition of distributions.
Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of distributions over $\C$.
Then $\struct {\map {\DD'} {\R^d}, +, \, \cdot \,}_\C$ is a vector space.
Proof
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Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples