The concept of "orthonormal basis" in my linear algebra textbook is introduced directly in the context of "vector space". But I think it needs the additional structure defined on the vector space. We cannot talk about "orthonormal" without inner-product. Am I right?
Further more, I cannot imagine the real usage of vector space without any additional structure: No norm, no inner-product, no length, no degree. We only have linear combination. Is there some important theorems on pure vector space or normed vector space? If no, why don't we define like "vector space must have inner-product"?
Best Answer
You are right, a vector space must also have an inner product before the concept of an orthonormal basis makes sense (and if the vector space is infinite dimensional there are other technical requirements as well - it needs to be either separable or complete). A vector space with an inner product is called an inner product space.
However, vector spaces without inner products are still useful in their own right. For example, the solutions to a system of linear equations form a vector space, as do the solutions to a system of linear differential equations.