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In my linear algebra course on the faculty of mathematics, we used the following notation for the inner product: $$\langle v, w \rangle$$ On the other hand, my friends from the faculty of physics rather use the following notation: $$(v \mid w )$$

Where do the two notations come from? Is there any reason why the former is more widespread on my faculty than the latter (and the same for the other notation). Are there any cases when one is more convenient than the other?

One case which comes to my mind, that that the former may be confused with $\langle v, w \rangle_\mathbb{K}$, which would be used for a linear span of these two vectors.

marmistrz
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    Are you sure the physicists are not writing $\langle v\mid w\rangle$? That would be bra-ket notation, which is popular in physics -- it was invented by Dirac, a physicist, for use in quantum mechanics. – hmakholm left over Monica May 14 '16 at 11:28
  • Notations for inner products are $(v,w)$, or $\langle v,w\rangle$, or $B(v,w)$. In the context of simple Lie algebras, $\langle v,w\rangle=\frac{2(v,w)}{(v,v)}$. – Dietrich Burde May 14 '16 at 11:44
  • I wrote it as I encountered it. The author of the script is a quantum field theory specialist, this may be of importance – marmistrz May 14 '16 at 12:13

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It's the same thing, except physicists uniformly make the first vector get the conjugate in the complex case and mathematicians often make the second one get the conjugate.

The physics people further like to separate the inner product $\langle v|w \rangle$ and consider $\langle v|$ and $|w\rangle$ separately, where $|w\rangle$ is the vector and $\langle v|$ is the linear functional formed by taking the adjoint (transpose-conjugate) of $|v\rangle$. This is called Dirac notation. A mathematician might write $\langle v|$ as $v^*$, which again confuses the physicists, because physicists use asterisks where mathematicians use conjugation bars.

LeopardShark
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B Chin
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  • It's mostly just old-school mathematicians that tend to conjugate the first argument, not most modern mathematicians, nor most others. "This is how the inner product was originally defined and is still used in some old-school math communities. ... engineering and computer science, and most of physics and modern mathematics ... for any inner product, there is some hermitian, positive-definite matrix M such that <x,y>=x^*My." Note 1 of https://en.wikipedia.org/wiki/Inner_product_space#Notes Cf. https://math.stackexchange.com/questions/129227/cross-product-in-complex-vector-spaces/3567679#3567679 – user3810316 Apr 23 '21 at 08:09