The very definition of an inner product

Question

Let's consider a finite, n-dimensional inner product space. My first question is, is the inner product between a pair of vectors $v_1 = \sum_{i=1}^n \alpha_ie_i$ and $v_2=\sum_{i=1}^n \beta_ie_i$ where $\{e_i\}_{i=1}^n$ is a basis, only defined when $\{e_i\}_{i=1}^n$ is orthogonal? In the case of $\{e_i\}_{i=1}^n$ being orthonormal, then $$(a,b) = \sum_{i=1}^n \beta^*_i\alpha_i$$ But what if $\{e_i\}_{i=1}^n$ is not orthogonal? Will the inner product between $a$ and $b$ be still given by the same above formula?

Answer 1

Before you have an inner product, you can't say that the standard one is orthogonal. Orthogonality acquired meaning in the presence of an inner product. You can define different inner products, and for each you will get different collections of orthogonal, or orthonormal, bases. Fixing an inner product, say on a real vector space, you then can easily show that for each orthonormal basis the inner product $\langle u, v\rangle$ is represented as $\sum u_kv_k$, where the $u_k,v_k$ are the coefficients of $u,v$, respectively, in the chosen orthonormal basis.

Answer 2

Your last comment shows you are becoming more proficient with the concept of an inner product.

Yes, the inner product can exist in many different forms -- e.g., in the form of an integral for polynomial function spaces. The dot product that you learn in Calc III is another specific example of an inner product.

You must be in an inner-product space first before talking about orthogonality.

Finally, inner products induce norms on vector spaces. You should check what this means.