The equivalent of a dot product in Hilbert space is:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \langle f | g \rangle = \int f(x) g(x) dx [/tex]

And you can find the angle between functions/vectors f and g via:

[tex] \theta = arccos\left( \frac{\langle f | g \rangle}{\sqrt{\langle f|f \rangle \langle g|g \rangle}} \right)[/tex]

So is it possible to come up with a cross product between two Hilbert space vectors? I know with discrete dimensionalities, you can only have vectors result from cross products in 3D and 7D, but is it possible here?

If not, is it possible to get something akin to a bivector, like with 4D cross products?

Is it possible to get something like:

[tex] \langle f | \times |g \rangle = | h \rangle [/tex]

where

[tex] \langle f |h \rangle = 0 [/tex]

[tex] \langle g |h \rangle = 0 [/tex]

Maybe the cross product is a more complicated object akin to a bivector, like something of the form:

[tex] \langle f | \times |g \rangle = | h \rangle | k \rangle [/tex]

or

[tex] \langle f | \times |g \rangle = | h \rangle \langle k | [/tex]

Is there a formal way to construct a cross product like this in Hilbert space?

As an aside: if

[tex] cos \theta = q [/tex]

then

[tex] sin \theta = \sqrt{1 - q^2} [/tex]

And for cross products, the maginitude is [tex] |A \times B| = |A||B|sin \theta [/tex]

So it seems to follow from the second equation in this post, that the magnitude of a Hilbert cross product is:

[tex] | \langle f | \times |g \rangle | = \sqrt{ \langle f|f \rangle \langle g|g \rangle - \langle f | g \rangle^2 } [/tex]

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# Hilbert Space Cross Products

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