Dear  Esakki ,

Please see the attachment.

Regards,

Wiwat Wanicharpichat

 Dear Esakki,

saying roughly, inner product generalizes the dot product to abstract vector spaces over a field.

In Euclean space inner product and dot product is the same, and very often called both dot and inner product.

Greetings, Tadeusz.

   Loosely speaking, an inner product is anything that takes two "vectors" from your space, and begets a number. For example, if your "vectors" are functions, the inner product can be an integral of their product . 

The dot product is for computing angles in a means consistent with Euclidean geometry. A general inner-product is just a means of extending this notion of angle to other spaces with Euclidean structure.

Thank you for all your answers. I did PhD in pure maths. I would like to work on Quantum gravity. Can I apply for another PhD for physics?

Often the requirement for a PhD is three publications, but only one unpublished paper from your hand is available here at ResearchGate. This looks strange and perhaps it will even make it more difficult for to get a second PhD. You have written that "I am not satisfied with my PhD." and now you ask if you can apply for another PhD. Do you want to take another PhD because you were not satisfied with the first? 

I have three publications. If you need I will send my soft copy of my thesis. I have written  some results which are unpublished. For eg, I uploaded only one research paper which was part of my thesis.

Why not? It is never too late.

The dot (inner) product is far more general than anyone has mentioned. In general the inner product is a binnary opperation on multivectors that produces a multivector of lower rank.

Vectors are rank 1 multivectors so their inner product is a rank 0 multivector ie a scalar.

The inner product of a vector and bivector (rank two multivector) is a vector

The inner product of two trivectors is a scalar  and so on

For details see David Hestines "New Foundations in classical mechanics"  and Doran and Lazenby "Geometrical Algebra for physicist." If I remember rightly the latter is published by Cambridge University Press

Esakki, if you are interested in quantum gravity, take a look at the papers on my RG page and tell me what you think.