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1

The nub of it is that total boundedness is an intrinsic property of a metric [more generally, uniform] space. It does not depend on whether the space is a subspace of some larger space, and if so, what that larger space is, all that matters is the metric. We have two metric spaces, $X = A^\ast(S^\ast)$, with the metric induced by the norm on $E^\ast$, and ...

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Wikipedia answers both of your questions. Specifically, let $X$ be a topological vector space and $M \subset X$ be a subspace. Then $X/M$ in its usual topology is Hausdorff if and only if $M$ is closed. This is essentially because a topological vector space is Hausdorff if and only if $\{ 0 \}$ is closed, and $\{ 0 \}$ is closed in $X/M$ if and only if $M$ ...

2

Suppose $f$ is as stated with exponential bound $Ce^{-\delta|x|}$, and suppose that $\overline{g} \in L^{2}(\mathbb{R})$ is orthogonal to the functions $\{ x^{n}f(x) \}_{n=0}^{\infty}$ in the $L^{2}$ inner-product. Define $$H(\lambda) = \int_{-\infty}^{\infty}g(x)f(x)e^{-i\lambda x}\,dx.$$ Suppose that $|\Im\lambda| \le \delta' < ... 3 These conditions are not equivalent. Take a non-zero operator$x$. Then certainly$x\neq -x$. However, $$|(-xh, k)| = |-(xh,h)| = |(xh, k)|$$ for any$h,k\in H$. Separation in the context of locally convex spaces means the following: Let$\mathscr{P}$be a family of seminorms. Then$\mathscr{P}$is (by definition) separating if for each$x\neq 0$... 2 No, they do mean that$X$is a countable [countably infinite] set, and the measure is the counting measure on$X$, $$\mu(A) = \operatorname{card} A$$ for any subset$A\subset X$. In particular, the$\sigma$-algebra of$\mu$-measurable sets is$\mathfrak{P}(X)$, the entire power set of$X$, and the only null set is$\varnothing$. Thus for$f\colon X \to ...

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Topology on $C_{\mathrm{compact}}^{\infty}(R)$, which is given by all the good semi-norms, is generated by the following collection of semi-norms $\| \cdot\|_{m,\epsilon}$ $m=\{m_i\}_{i=1}^{\infty}$ is an increasing positive integer sequence, $\epsilon=\{\epsilon_i\}_{i=1}^{\infty}$ is a sequence converging to zero from above. \$K_{i}=[-i,+i],\quad ...

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