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You really only need to consider your first equation in order to obtain a contradiction. Suppose that there exists $w \in L^{1}_{\mbox{loc}}(\mathbb{R})$ such that $$\int w\varphi dx = \varphi(0),\;\;\;\varphi \in \mathcal{C}_{c}^{\infty}(\mathbb{R}).$$ That equation is inconsistent. To see why, find ...

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$\underline{Hint}$: To show that $\{ e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis (ONB) for $L^2(\mathbb{T})$, we must show that $\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}$ is dense in $^2(\mathbb{T})$, which is equivalent to stating that $\exists g\in \operatorname{span}\{ e_n\mid n\in\mathbb{Z}\}$ such that  \|f-g\|_{L^2(\mathbb{T})}=\left( ...

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The extended norm makes your vector space into a topological group, that is, addition is continuous. But multiplication $\mathbb R \times V$, $(\lambda,(x_n)_n)\mapsto (\lambda x_n)_n$ fails to be continuous. Nevertheless, the uniformity given by $\|\cdot\|$ is complete: If $x^N$ is a Cauchy-sequence in $V$ then, for some $N_0$, the sequence ...

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