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My question is an extension to the post How is the acting of $H^{-1}$ on $H^1_0$ defined?. Here duality pairings were discussed and even given explicit examples.

Let $U$ and $V$ be Hilbert spaces such that $U\subset V$, with inner products $(\cdot , \cdot)_{U}$, $(\cdot , \cdot)_{V}$. Can we give explicit examples to understand the difference between dual pairings $\langle\cdot , \cdot\rangle_{U}$, $\langle\cdot , \cdot\rangle_{V}$ and inner products? and when is it true that for $u\in U$ and $v\in V$

$$ \langle u,v\rangle_U = (u,v)_U $$

Thanks in advance.

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