I have some problem in the definition of inner product space. The book I use to learn in linear algebra and its application 4th edition (David C.Lay)
In the chapter 6.7 it define the inner product space is an vetor space with an inner product. Also, in its example the inner product of the vetor $u=(u_1,u_2)$ and $v=(v_1,v_2)$ is set that $$<u,v>=4u_1v_1 + 5u_2v_2$$ Obviously, this satisfy all the axioms of an inner product. However in it previous chapter (chap 6.1) it define an inner product of 2 vector u and v to be $$<u,v>=u_1v_1 + u_2v_2$$ That's the point I don't understand. For me the latter make sense while the former seems a bit confusing. Does the inner product of the two vector u and v is defined by its entries $(u_1,u_2)$ and $(v_1,v_2)$ but why the inner product of the former is $4u_1v_1 + 5u_2v_2$.
Moreover, the definition of an inner product space is a vector space with an inner product. Does this mean that $R^n$ with any real number (inner product) can be called an inner product space. For example: $R^n$ and 3 is an inner product space ?
Can anyone explain it more specific with some detailed example