Inner product vs dot/scalar product

[cianfa72]

TL;DR

About the difference between inner product and dot/scalar product

Hi,

from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things.

Given a vector space $V$ an inner product $\langle . | . \rangle$ is defined between elements (i.e. vectors) of the vector space $V$ itself. Differently dot/scalar product $\cdot$ is defined between an element of the vector space $V$ and an element of the dual vector space $V^*$.

Then given the inner product in $V$ there is a canonical isomorphism between $V$ and $V^*$ hence the result of the inner product between two vectors vs. the scalar product between the first vector and the dual vector canonically associated to the second vector is actually the same.

Does it make sense ? Thanks.

 

[robphy]

TL;DR Summary: About the difference between inner product and dot/scalar product

from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things.

Since your question is about terminology,
I suggest that you provide "quotes" of the definitions from the Penrose book.

 

[cianfa72]

Here Penrose defines the scalar product between a covector $\alpha$ and a vector $\xi$.

 

[martinbn]

The same term can mean different things in different contexts. Scalar product can be used (and is used) to mean the same thing as inner product. It can also be used to mean the canonical pairing between vectors and covectors.

 

[Svein]

There are more "inner products" than scalar products. Check out Hilbert Space definition.