# The Unapologetic Mathematician

## Dirac Notation II

We continue discussing Dirac notation by bringing up the inner product. To this point, our notation applies to any vector space and its dual, with the ket $\lvert v\rangle$ denoting a vector $v\in V$ and the bra $\langle\lambda\rvert$ denoting a linear functional $\lambda\in V^*$. The evaluation $\lambda(v)$ is then denoted by the bra-ket pairing $\langle\lambda\vert v\rangle$.

But the neat thing about this notation is that it makes bras look like some sort of reflection of kets. And they are, in a sense. The dual space $V^*$ is some sort of reflection of the vector space $V$, but there’s no clear mapping from vectors in one space to vectors in the other; unless, that is, we pick a specific isomorphism; or, equivalently, an inner product.

When we’ve got an inner product in the picture, we get a (conjugate) linear isomorphism that sends the vector $v$ to the linear functional $\langle v,\underline{\hphantom{X}}\rangle$. In Dirac notation, we send the ket $\lvert v\rangle$ to the bra $\langle v\rvert$. Then the value of this linear functional on a vector $w$ (the ket $\lvert w\rangle$) is the pairing $\langle v\vert w\rangle=\langle v,w\rangle$, just as it should be.

July 1, 2009 - Posted by | Algebra, Linear Algebra

## 6 Comments »

1. Assuming we’re working over the complex numbers, the map sending v to the linear functional is surely conjugate-linear, not linear? (Apologies for lack of LaTeX!)

Comment by Matt Daws | July 1, 2009 | Reply

2. Yes, I’ve mentioned it explicitly before, but I should still make it explicit here.

Comment by John Armstrong | July 1, 2009 | Reply

3. Have you looked at rigged Hilbert spaces at all? They made taking QM from physicists much more tolerable. =)

Comment by Douglas | July 2, 2009 | Reply

4. Yes, I’ve looked at them, Douglas, but I’m not sure why you bring them up in the context of a purely notational discussion.

Comment by John Armstrong | July 2, 2009 | Reply

5. […] Notation III So we’ve got Dirac notation and it’s nice for inner product spaces, but remember we’re not just interested in vectors and vector spaces. We’re even more […]

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6. […] and Bilinear Forms on Inner Product Spaces in Dirac Notation Now, armed with Dirac notation, we can come back and reconsider matrices and forms. For our background, we’ve […]

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