I Inner product vs dot/scalar product

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Inner products and dot/scalar products are distinct concepts, as clarified by Penrose in "The Road to Reality." An inner product is defined between vectors in a vector space, while a dot/scalar product involves a vector and an element from its dual space. There exists a canonical isomorphism between the vector space and its dual, making the results of the inner product and the scalar product equivalent in certain contexts. Terminology can vary, with "scalar product" sometimes referring to the same concept as "inner product" or to the pairing between vectors and covectors. Understanding these distinctions is crucial for clarity in mathematical discussions.
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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.
 
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cianfa72 said:
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.
 
Capture.JPG


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