What does the dot product in Hilbert Space suggest?
When you take the dot product of two vectors in an ordinary (n-dimensional) Cartesian space, you are measuring the length of the projection of one vector along the direction of the other. This is equivalent to measuring the cosine of the angle between them. If the scalar product is zero, the two vectors are orthogonal. That is, the projection of one vector on the other has zero length. In an abstract space like a Hilbert space, the scalar product is a measure of the “overlap” of two of the “vectors” in the space. Typically in a problem described by a Hilbert space, we decompose the space into an orthonormal basis, which is a set of functions (vectors in the space), that are orthogonal in the sense that their scalar product (however that’s defined for the space) is zero. Given an arbitrary function, one can find it’s representation in terms of the orthonormal basis by computing the scalar product with each of the basis functions in turn. Those scalar products are the coefficients in the
