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As discussed in this question, any inner product space $(V, \langle \cdot, \cdot \rangle)$ can be considered as self-dual and so a "dual pairing" can be effected via the inner product, e.g., $$ \langle u, v \rangle $$ Typically however once considers dual pairings between a vector $v \in V$ and a functional $\phi \in V^*$, e.g., $$ (\phi, v) := \phi(v) $$ I have often seen this latter type of pairing denoted by $$ \langle \phi, v \rangle := \phi(v) $$ and even by $$ ( \phi | v ) := \phi(v) $$ The first variant obviously conflicts with the customary notation for the inner product and I would rather avoid it. So, what is the best/customary way to notate the dual pairing in this sense? Would it be correct/meaningful to use the dirac notation and write $$ \langle \phi | v \rangle := \phi (v)? $$

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Yes, for example Dirac's bra-ket notation is employed effectively (and quite elegantly) in various expositions of the Umbral Calculus - an effective calculus of adjoints that serves to systematically derive and classify almost all of the classical combinatorial identities for polynomial sequences (e.g the sequences of Abel, Appel, Bell, Bernoulli, Bessel, Boole, Boas-Buck, Euler, Gould, Hermite, Laguerre, Mahler, Meixner, Mittag-Leffler, Mott, Poisson-Charlier, Sheffer, Stirling, etc), along with associated identies (generating functions, expansions, duplication formulas, recurrences. inversions, Rodrigues formula, etc, e.g. the Euler-Maclaurin expansion, Boole's Summation formula, Newton interpolation, Gregory integration, Vandermonde convolution). For example almost all of the identities in Riordan's classic book Combinatorial Identities can be systematically derived and classified via the Umbral Calculus. For a very nice introduction see Steven Roman's book The Umbral Calculus.

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Thanks for the references and verifying that the Dirac notation I indicated would work an this instance. I thought that it would apply, but wasn't 100% certain. – ItsNotObvious Nov 30 '11 at 17:40

I don't think you can depend on a distinction between $({\cdot}\mid{\cdot})$, $\langle{\cdot}\mid{\cdot}\rangle$, and $\langle{\cdot},{\cdot}\rangle$ being noticed by the average reader without copious explicit warnings. But does it matter? I.e., is there a real possibility of confusion?

Formally, you could even declare that $\langle{\cdot},{\cdot}\rangle$ only works for $V^*\times V$. Then equipping $V$ with an inner product really means choosing to identify $V$ with a (not necessarily proper) subspace of $V^*$, and similarly working in a dual pairing between $W$ and $V$ means working with an invisible linear map $W\to V^*$. Unless $W$ happens to intersect $V$, there should be no risk of confusion.

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The issue I have with all of your suggestions is that they do not visually distinguish between elements of $V$ and elements of $V^{\ast}$.

Personally I use notation like $v, w$ to denote elements of a vector space $V$ and notation like $v^{\ast}, w^{\ast}$ to denote elements of the dual space $V^{\ast}$; then I just write the dual pairing as $v^{\ast}(w)$ or similar. Another notation which makes this distinction is Einstein notation, although I personally dislike it because it relies on an implicit choice of basis.

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