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I am slightly confused about the distinction and definition of duality and natural pairings. I think one is defined as:

$$ \langle f,v\rangle=f(v)$$

but I am unsure which one and how the other is defined. Please can someone explain? (the information on wikipedia does not seem to agree with other sources).

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  • $\begingroup$ What i know (i'm not an expert in topology and functional analysis) is that they are often synonymous. If not, then the natural pairing is defined for a vector space and the dual space. The dual pairing is more general defined for two vector spaces. $\endgroup$ Commented Jul 16, 2017 at 9:25
  • $\begingroup$ @Fakemistake this is what wikipedia says but I can't find it in any other source. $\endgroup$ Commented Jul 16, 2017 at 9:27

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Given any vector space $X$, one can consider the algebraic dual space $X'$ of all linear functionals (there is no notion of continuity, yet). From this we can form a pairing $\langle x,f\rangle$ defined by $\langle x,f\rangle := f(x)$ ($x\in X, f\in X'$). However, these are not the only kind of pairings that can be formed.

We could instead replace $X'$ with any other vector space $Y$ for which we can form a bilinear mapping $\langle \cdot,\cdot\rangle:X\times Y\to \Bbb F$ (where $\Bbb F$ is the underlying field) and $Y$ contains "sufficiently many" vectors to separate the points of $X$. Such a pairing is perfectly well defined, but is not as "natural" as a pairing between $X$ and $X'$. (I use the word "natural" with caution here.)

If the vector space $X$ has topological structure, for example $X$ is a Banach space, then we can instead consider the pairing between $X$ and its topological dual $X^*$, again defined by $\langle x,f\rangle := f(x)$ ($x\in X,f\in X^*$). This is the obvious choice of pairing for Banach spaces. Here what is happening is we are considering the pairing between $X$ and the subspace $X^*\subseteq X'$.

If, however, $X$ is merely a topological vector space, then $X^*$ might be a trivial vector space, but we can still form a "pairing" - not that it is particularly interesting.

If $X$ is a Hilbert space, then there is another pairing that we might consider: the one given by the inner product on $X$. In fact, this turns out (via the representation theorem) to be the "same" as pairing between $X$ and its topological dual space (ignoring the need to take complex conjugates if $\Bbb F = \Bbb C$ for a moment).

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