[Solution Library] Subspaces. Consider the space F of all functions f: R_+ \rightarrow R, which have a Laplace transform f#770;(s)=∫_0^∞ f(t) e^-s
Question: Subspaces. Consider the space \(F\) of all functions \(f: \mathbb{R}_{+} \rightarrow \mathbb{R}\), which have a Laplace transform \(\hat{f}(s)=\int_{0}^{\infty} f(t) e^{-s t} d t\) defined for all \(\operatorname{Re}(s)>0\). For some fixed \(s_{0}\) in the right half-plane, is \(\left\{f \mid \hat{f}\left(s_{0}\right)=0\right\}\) a subspace of \(F\) ?
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