I searched the forums, and I can't find anywhere where someone asked this question point blank. I may be completely off base so I apologize in advance if that is the case. In quantum mechanics, an inner product is formed as the bra-ket <φ|ψ>. We are told that vectors are represented by the kets and the bras represent dual vectors. However, after reading linear algebra texts, I understand that inner products are not generally formed from a vector and a dual vector, but from two vectors in the same space. Now let V, V* and F be a vector space, dual space, and field respectively. I am familiar with the Riesz representation theorem used to justify the bra-ket formalism, however the fact that the dual basis is defined as ei(ej) = δij and the fact that the inner product is a linear functional mapping elements in V to the field F are very suggestive that a natural way to define the inner product should be between a vector in V and a dual vector in V*. Also, the use of the Einstein summation convention in relativity is suggestive because it sums over a vector with an “upstairs” index and a vector with a “downstairs” index. If two vectors with an “upstairs” index are to be summed over, one has to use the metric to lower the indices of one of the vectors. Consider a vector space over the reals. Yet another example would be a representation of a vector as a column vector. By the ordinary rules of matrix multiplication, a scalar product can only be formed with a row vector. Often column vectors are identified as vectors and row vectors as dual vectors. My question is why are inner products defined in linear algebra texts as between vectors in the same space instead of between a vector and a dual vector when in practice it seems the opposite is true? Is there a reason why the inner product is not defined as between a vector and a dual vector? Why is the inner product defined as between two vectors of the same space? Given the Riesz representation theorem, perhaps my questions are moot, but I would still like to conceptually understand this.