In Normed linear space, we study about distance only. But in dot
product we can talk about angles also.when we generalize the notion
dot product we get inner product for any dimension.
Because the inner product (specially the Hilbert) spaces provide a generalization (infinite dimensional) of the Euclidean spaces Rn. Nevertheless, the geometrical meaning is not the same. This is supported by the Schwarz inequality.
Dear Esakki,
As Rogerio alluded to in his post, the inner product generalizes ideas from Euclidian spaces. Perhaps a concrete example might help, suppose we have two vectors t and u and we would like to find the shortest distance from t to some point in the direction u. Then we can work out via Pythagoras that relation in R^2, and by extension, we can find the same relation in R^n. This is such a fundamental operation that we might think about what other ways to measure might preserve this structure, and as you can work out, it will turn out to be an inner product. Now with this more general object, functions can be thought of as vectors and inner products can become integrals and we are able to do all this thanks to the generalization that Rogerio pointed out in his post. Hope this helps,
Alfonso
I Euclidean space of dimension 2 and 3 angles play an important role and can calculated from the dot product. The angle between two vectors can be defined for vectors in a Hilbert space, but angles are not used much in higher dimensions except right angles that correspond to the dot product being zero.
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?
This depends on the institution. In most cases the prerequisite for PhD. study is a university education in the same or similar field.
Why do you want to become Post. Doc. when you are assistant professor?
@Prof.Peter
I am not satisfied with my PhD. I want to learn more. I need mentor and Library. At Present I don't have enough time to think and do research.
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics