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In Stochastic Partial Differential Equations by Lototsky & Rozovsky (2017) it says the following:

Let $V$, $H$, and $V’$ be three Hilbert spaces. We say that $(V,H,V’)$ is a normal triple if:

  1. $V$ is a dense subspace of $H$, and the inclusion $V \hookrightarrow H$ is continuous;
  2. $H$ is a dense subspace of $V’$, and the inclusion $H \hookrightarrow V’$ is continuous;
  3. The following inequality holds: $$ |\langle h, v \rangle_H| \leq |h|_{V’} |v|_V \quad \text{for all } h \in H,; v\in V. $$

Importantly, they emphasize:

"We do not assume that $V'$ is the dual of $V$; the Hilbert space $V'$ is simply another space in the triple."
(Definition 2.1, p. 11)

However, they then introduce a bracket $[y, v]$ for $y \in V'$, $v \in V$, and define it via the limit

$[y, v] = \lim_{n \to \infty} \langle h_n, v \rangle_H,$

where $h_n \in H$ and $h_n \to y$ in the topology of $V'$.

My confusion is the following: This expression resembles a duality pairing between $V'$ and $V$. But the authors explicitly mention that $V'$ is not defined as the dual of $V$. There is no derivation of this pairing or any assumption that $V' \subset V^*$ or that $V‘$ consists of linear functionals.

So my questions is: How can the expression $[y,v]$ be justified if no duality is assumed between $V$ and $V'$?

Any help is greatly appreciated.

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    $\begingroup$ $h_n$ is an element of $H$, and $v\in V$, so it can be injected to give $\iota_{V\to H}(v)\in H$. We can of course take the inner product of these two guys in $H$, so $\langle h_n, \iota_{V\to H}(v)\rangle_H$ is a well-defined quantity. So I don’t seee any issues here. The only thing to address is why the limit as $n\to\infty$ exists (show it is a Cauchy sequence of complex numbers), and why it is independent of the sequence $h_n$ used to approximate $y\in V’$ (I’d rather call it $\tilde{V}$ at this stage); these are both easy enough statements so I don’t see your issue with the definition. $\endgroup$ Commented Jul 7 at 7:08
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    $\begingroup$ the symbol $[y,v]$ is just that: a symbol. Its definition occurs on the RHS; your task is to simply show its well-definition by addressing the two points above. $\endgroup$ Commented Jul 7 at 7:10
  • $\begingroup$ @peek-a-boo Thank you. You are right the properties are enough to easily show that $[\cdot,\cdot]$ is a well defined bilinear form on $V\times V’$. I think my confusion arose from the missing assumptions on the structure of $V’$. Especially since I thought of it a version of a Gelfand triple, while I should have viewed the Gelfand triple as a version of a normal triple. Do you know by any chance why Lototsky & Rozovsky consider normal triples instead of Gelfand triples, while this seems rather unusual? $\endgroup$ Commented Jul 7 at 15:46
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    $\begingroup$ Probably because the pairing allows one to identify the (topological) dual of the left space with the space on the right. I haven’t checked in detail if this is true (one has to be mindful of which norms are used… since for example, point 3 is not simply an application of Cauchy-Schwarz and continuity of the inclusions). $\endgroup$ Commented Jul 7 at 17:14

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