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3

This is a general fact about seminormed spaces. You have a family of seminorms $\|\cdot\|_{\alpha,\beta}$. The sets $U_{\alpha,\beta,r} = \{\phi : \|\phi\|_{\alpha,\beta}<r\}$ form a sub-basis of neighborhoods of zero, meaning that their finite intersections yield an basis of such neighborhoods. This is precisely the topology for which the convergence ...


2

For $(1)$, to prove that $\mathcal D(U)$ with $\{\| \cdot \|\}_{m \in \mathbb N}$ is a locally convex topological vector space, you need to show that $\{\|\cdot\|_m\}_{m \in \mathbb N}$ is a separating family of semi-norms on $\mathcal D(U)$. The question already says "norms" in it, so maybe you just have to show that it's separating, i.e., given $\phi \in ...


1

Assuming we've shown that $\mu$ is a measure, and that we're only concerned with bounded sets: Say $B$ is a bounded closed set and $E\subset B$ is measurable. First, note that $$\mu(E)=\inf\{\mu(V):E\subset V\subset B, V\text{ relatively open}\}.$$ (Here "relatively open" means open relative to $B$; that is, $V=B\cap W$ where $W$ is an open subset of $\Bbb ...


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A more general theorem in Folland's Real Analysis can answer this question:


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This is indeed correct. You might want to expand on why these (basic) open sets are indeed disjoint (triangle inequality argument). Abstractly, this argument works because the point-evaluations $e_v$, where $v \in V$, form a family of functions (to a Hausdorff space, namely the underlying field) that separates points, in this case basically by definition ...


1

Hint: ${s\over{s+t}}x+{t\over{s+t}}y\in A$ if $x,y\in A$ and $A$ is convex, thus $(s+t)({s\over{s+t}}x+{t\over{s+t}}y)=sx+ty\in (s+t)A$


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My question is: if I permit partially ordered countable families of Fréchet spaces, but still require the inductive limit to be strict (as defined above), do I wind up with the same class of LF-spaces, or is this definition actually more general? Fact: The limit $E$ of a countable inductive system $(E_i)_{i \in I}$ of Fréchet spaces is already an ...



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