Now, in classical mechanics we deal with a vector space called Euclidean space (or Cartesian space), which is defined
by the unitary vector components ex, ey and ez extending in a particular direction to infinity. Therefore this space
is entirely filled by the vector e.
Hilbert space is, by contrast, a much more generalised and flexible version of this space. In Hilbert space, an
“axis” can be any function or vector (not only e), which can extend for any distance or to infinity.
Imagine a space containing a number of representations completely orthogonal (i.e. at π∕2 radians, or
90°) to one another which are each represented by a different basis function or vector. Each can have vectors projected on to them to reveal a different property of a corresponding vector or function. Suppose we have the basis functions ψifor i = 1 to 7 in this Hilbert space.
Notice how each basis vector is linearly dependent of every other basis vector within its set, meaning they are orthogonal and have an inner product of 0 with respect to one another.
Let’s now introduce into it a vector ψ, which has projections onto (or “components in”) each of the basis vectors ψi given by
As we can see, we’ve introduced this new vector in red and ‘visualised’ projecting it onto each of its basis vectors. We’ve previously discussed the similarities between Euclidean space and Hilbert space and can here draw a parallel between the process of resolving vectors on to their Cartesian axes and the projection of a state onto its basis.