Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##.(adsbygoogle = window.adsbygoogle || []).push({});

Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##.

In Dirac's Bra-Ket notation, this is written as, ##u\otimes v:=\left|u\right>\otimes \left| v\right>:=\left|u\right>\left| v\right>##

What is the outer product, then?

In Dirac notation, it is written as ##\left|u\right> \left< v\right|##. Which makes it clear that the outer product is a linear functional acting on some ##v'\epsilon ~V## and giving some ##u'\epsilon~ U##, such that ##\left| u'\right>=\left<v|v'\right>\left|v\right>##.

So, would it be correct to say that (##\otimes_O## is the outer product)

$$u\otimes _Ov:=u\otimes v^* $$

for, ##u,~ v,~v^*~\epsilon ~U, V, V^*##, respectively?

Also, would it be correct to say that ##u\otimes v## is a type ##(2,0)## tensor while, ##u\otimes _O v## is a type ##(1,1)## tensor?

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# A Difference Between Outer and Tensor

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