> Not all vector spaces are equipped with an inner product.
Any vector space over a field (usually part of the definition of a vector space) is equipped with the standard inner product, because multiplication and addition are part of the definition of a field.
That’s not how it works. Unless you explicitly give the vector space an inner product, it doesn’t have one.
What you probably mean is that “you can always define an inner product” but that’s a very different statement.
That’s not true either though, the different dimensions in a vector space do bot have to belong to the same field, so you can’t assume you can add them together.
> Unless you explicitly give the vector space an inner product, it doesn’t have one.
Well, no, not at all.
The inner product is still there. It's still an inner product. The space in which your vectors exist is still an inner product space. You may not care about the inner product, but it doesn't cease to exist when you stop looking at it.
I really don't understand why you would say this, it's obviously false.
Setting aside the subtler point of what it means to "have" something in mathematics:
clearly only some vector spaces even have the potential to introduce an inner product. Consider F for some random finite field. You can make a vector space from it, but what would the inner product be? Or R x F for that matter, you could never give that an inner product.
That's why the concepts of "vector space" and "inner product space" are separate concepts. Some vector spaces aren't, and could never be, inner product spaces.
Any vector space over a field (usually part of the definition of a vector space) is equipped with the standard inner product, because multiplication and addition are part of the definition of a field.