Measurement and duality mentioned in Lecture 4 \(k\)-forms of the CMU Discrete Differential Geometry course is enlightening. Inner product of two vectors \(u\) and \(v\) is just a kind of such measurement, which is intrinsically a projection operation. \(v\) is the vector to be measured and \(u\) is the vector for measuring. Even though \(u\) and \(v\) are both vectors in an Euclidean space, they have different identities. That’s why during the evaluation of \((u, v)\), \(u\) is written as a row vector, while \(v\) is a column vector. Similar cases arise in other areas, such as in functional analysis, a vector in a Hilbert space is associated with its dual vector via the Riesz representation theorem; in quantum mechanics, there is the demarcation of ket \(\langle\psi \vert\) and bra \(\vert\varphi\rangle\); in PDE or FEM, ansatz and test basis functions are adopted to discretize bilinear forms.

Backlinks: 《Concept of duality》