I am taking my first steps in learning quantum mechanics and am learning about Dirac's bra-ket notation. I am trying to understand what the inner product is.
My understanding so far: the inner product is an operation between to vectors which returns a scalar. This allows us to define orthogonality: two vectors are orthogonal when their inner product is 0. The inner product is a generalisation of the dot product I have been using up till now, which is essentially the inner product restricted to vectors in $\mathbb{R}^{n}$, always returning real scalers. An inner product space is a vector space for which the inner product is defined.
This is where I get confused: so far, I have been applying the dot product to vectors from the same vector space. Furthermore, from Wikipedia: the inner product "associates each pair of vectors in the [inner product] space with a scalar quantity known as the inner product of the vectors."
However, going through Shankar's Principles of Quantum Mechanics, I have learnt that the vector space of kets has an associated vector space of bras, its dual space. The textbook states the inner product is only defined between bras and kets and hence only between a vector space and its dual space. I haven't found anything about an inner product kets or bras, and my gut feeling is it wouldn't make sense. Are the associated vector spaces to bras and kets not inner product spaces? Or would the inner product just be meaningless?
In summary, is the inner product within a vector space the same as the inner product between bras and kets, or am I confusing two different ideas? What are, in general, the operands on which the inner product acts?