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I am reading Ballentine's Quantum Mechanics and, in the mathematical preqrequisites, he asserts that, by construction, the Riesz Representation Theorem establishes an antilinear correspondence between bras and kets in the sense that, if $\left|F \right\rangle$ has as its dual $\langle F |$, then $c_1 \left|F_1 \right\rangle+ c_2 \left|F_2 \right\rangle$ is dual to $c_1^* \langle F_1 | + c_2^* \langle F_2 |.$

In previous QM courses, this has been laid down as an axiom, so I am hoping someone can elucidate for me how it follows by construction from the Riesz Representation Theorem, as given by Ballentine:

enter image description here

It must have something to do with the line below "Now construct the following vector:, but I can't quite put my finger on it.

EDIT: It seems to me that it follows from our choice to make the first slot of the inner product antilinear. I guess hazy terms like "correspondence" just confuse me though.

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    $\begingroup$ Hello! It is preferable to type out screenshots or images of text; for formulae, one can use MathJax. Thanks! $\endgroup$
    – jng224
    Commented Sep 8, 2021 at 18:27

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I'm not totally sure what the question is. If you hand me a vector $f$ I can hand you a continuous linear functional $F : \phi \mapsto \langle f,\phi\rangle$. If you hand me a continuous linear functional $F$, then I can hand you a vector $f = \sum_i \underbrace{\overline{F(\varphi_i)} }_{\in \mathbb C}\ \varphi_i$, where $\{\varphi_i\}$ is any orthonormal basis of the Hilbert space and the line denotes complex conjugation.

Note that the fact that this indeed a vector in the Hilbert space is trivial for finite-dimensional spaces, and follows from the fact that all continuous linear functionals are bounded for infinite-dimensional spaces.

One can show without too much difficulty that this correspondence amounts to an isomorphism - a one-to-one pairing between vectors and continuous linear functionals. This pairing is antilinear in the following sense: if $f$ and $g$ are vectors, $F$ and $G$ the corresponding continuous linear functionals, and $\alpha\in \mathbb C$ then $$f + g \iff F + G \qquad \text{and} \qquad \alpha f \iff \overline\alpha F$$

Both statements can be proven straightforwardly from the pairing defined above.


It seems to me that it follows from our choice to make the first slot of the inner product antilinear.

Yes, that's right. Indeed, we can go the other way: given a vector $f$ we can produce a continuous, antilinear functional $\widetilde F:\phi \mapsto \langle \phi,f\rangle$, and given a continuous antilinear functional $\widetilde F$ we can produce a vector $f = \sum_i \widetilde F(\varphi_i) \varphi_i$. This correspondence is then linear in the sense that $$f + g \iff \widetilde F + \widetilde G\qquad \text{and}\qquad \alpha f \iff \alpha \widetilde F$$

Having built up all this structure, given a vector $f$ we use the shorthand $F \equiv \langle f|$ for its linear functional partner, and $\widetilde F \equiv |f\rangle$ for its antilinear functional partner, while defining $\langle f|g\rangle := \langle f,g\rangle$. This provides the rigorous underpinning of the bra-ket notation which is in universal use among working physicists.

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  • $\begingroup$ Thanks for this superb answer! If I may, I should just note that (I believe) "orthonormal" should be added before "basis" in your third line. $\endgroup$
    – EE18
    Commented Sep 8, 2021 at 19:33
  • $\begingroup$ @1729_SR Yep, you're right. Thanks. $\endgroup$
    – J. Murray
    Commented Sep 8, 2021 at 19:35

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