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OK, here is an example of what one has to expect: the two Banach spaces of convergent sequences $c$ and of square summable sequences $\ell^2$ are Banach spaces ($\ell^2$ is even Hilbert. As vector spa …

To get an example: consider non-negative smooth functions $\chi_n$ with compact support in the intervals $[0, 1/n]$ with maximum $1$ at some point in this interval. Note that such functions indeed exi …

The argument of BS works also in the case where $X$ is a Hausdorff locally convex space since the topological dual still separates points (by Hahn-Banach). This is enough to show that the (continuous) …

Many points have been mentioned, but scanning through old questions I found this here: nobody mentioned that one can classify Hilbert spaces so easily via the size of the Hilbert basis. If you would o …

Your question also makes sense for a Banach space and that is where the following results also apply. The Hilbert space case is then only a particular case where you can get some more information abou …

The classical Borel Lemma states that for an arbitrary sequence $(v_n)_{n \in \mathbb{N}_0}$ of complex numbers there is a smooth function $f\colon \mathbb{R} \longrightarrow \mathbb{C}$ with Taylor c …

Suppose for simplicity that $A$ and $B$ are unital. If now $\pi = \pi_1 \otimes \pi_2$ then the operators $\pi(a \otimes 1)$ commute with all the operators of the form $id \otimes B$ with $B \in B(H_2 …

Let me give you a counter-example with an associative $\beta$, i.e. a Frechet algebra for which the exponentials do not exist in general: The main reason is that a Frechet algebra needs not to be loca …

Yes, an $n$ positive map is also $n-1$ positive. Hence you map is $n$-positive for all $n$, i.e. completely positive. For a proof, you include $M_{n-1}(A)$ as upper left block into $M_n(A)$.

Maybe not in a single theorem, but you can go for Cor1 in Section 33 and Cor3 in Section 50 in Treves book.

This is probably not quite an answer to your question but rather a hint how a generalization should look like. As already mentioned in the comments, the discrete and continuous measure-theoretic $\ell …

If $\mathcal{A}$ is a complete locally convex (Hausdorff) associative unital algebra (over $\mathbb{C}$) one is interested in defining "transcendental" functions of a given algebra element $a \in \mat …

The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a Bana …

OK, on request, the following reference as answer: Pietsch, A.: Nuclear locally convex spaces, vol. 66 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York, Heidelberg, 1972 …

Well, that depends on what topology you want to put on the space of distributions. The weak$^*$ is probably not really the one you would like to take. Instead, the strong dual might be more useful. Th …