# Thread: [SOLVED] Inner product question

1. ## [SOLVED] Inner product question

I've just learned we can define an inner product "almost" as we want. I mean by that that it must satisfy the four conditions to be an inner product. My question is : How many different inner products can we define? Is there an infinity of them, or are there a finite number? If there is a finite number of them, how many are them?

2. Hi,

Originally Posted by arbolis
I've just learned we can define an inner product "almost" as we want. I mean by that that it must satisfy the four conditions to be an inner product. My question is : How many different inner products can we define? Is there an infinity of them, or are there a finite number? If there is a finite number of them, how many are them?
I can't prove it, but I'd say there is an infinity of them...

http://www.math.jussieu.fr/%7Enekovar/co/q/quad.pdf << page 31, exercise 2.3.9 at the bottom of the page. And in class, we encountered several inner products, based on quadratic forms with various coefficients...
(you can find a French version of this in the same site )

3. Originally Posted by arbolis
I've just learned we can define an inner product "almost" as we want. I mean by that that it must satisfy the four conditions to be an inner product. My question is : How many different inner products can we define? Is there an infinity of them, or are there a finite number? If there is a finite number of them, how many are them?
Well, if you consider the Euclidean space $\mathbb{R}^n$, we can define an inner product between two vectors $\textbf{u} = \left(u_1,\;u_2,\;\dots,\;u_n\right)$ and $\textbf{v} = \left(v_1,\;v_2,\;\dots,\;v_n\right)$ as

$\langle\textbf{u},\;\textbf{v}\rangle = a_1u_1v_1 + a_2u_2v_2 + a_3u_3v_3 + \cdots + a_nu_nv_n$ for any $a_i>0\in\mathbb{R}$ (try checking this).

That already gives us an infinite amount (by varying our choices of the $a_i$s), and there are also infinitely many other vector spaces for which we can define inner products.

4. Thanks to both. I also have another question : the elements of a vector space are called "vectors", right? Also, a matrix, a polynomial can be elements of a vector space. This implies that a matrix and polynomials can be called "vectors"?

5. Originally Posted by arbolis
Thanks to both. I also have another question : the elements of a vector space are called "vectors", right? Also, a matrix, a polynomial can be elements of a vector space. This implies that a matrix and polynomials can be called "vectors"?
Yes. By definition, a vector is an element of a vector space, so a matrix or a polynomial can indeed be considered a vector. However, outside of the context of general vector spaces, "vector" usually refers to a vector in Euclidean $n$-space, so be clear what you mean when you refer to vectors in this way.