Assume you have a separable inner product space $V$ with orthonormal basis $B=(e_a)_{a∈A}$ and an element $v∈V$. Then there is a simple algorithm to compute the representation of $V$ with respect to $B$: since $|v⟩ = ∑_{a∈A}⟨eₐ∣v⟩|eₐ⟩$ we simply need to do:
def coeff(v, B):
for eₐ in B:
yield inner(v, eₐ)
This allows us to do a peculiar thing: compute the representation of $v$ with respect to $B$, in a manner agnostic of the representation of $v$, using only inner product space operations. It seems to me the same is not possible in general vector spaces, but I am struggling to coming up with an ansatz to prove it; so here is my question.
Assume $V$ is a Vector space without any additional norm/inner product structure, $v∈V$ and $B=(e_a)_{a∈A}$ is a Hamel basis.
- Is there an algorithm, using only vector space operations, that produces the representation of $v$ with respect to $B$?
- What about the finite dimensional case?
NOTE: We may assume that $v$ is given in some other basis $B'$, but we may not assume that the representation of the basis vectors of $B'$ is known in terms of $B$. Otherwise, this would be a straightforward application of change of basis formula, and miss the point about trying to do it in a manner agnostic to the representation of $v$.
For a vector space $V$ over field $K$ we shall allow the field operations $+ₖ$, $⋅ₖ$, $-ₖ$, ${()^{-1}}ₖ$, the vector operations $+ᵥ$, $-ᵥ$, the scalar product $⋅$, as well as equality checks $=ₖ, =ᵥ$.
Observations:
- In the case of a finite dimensional VS over a finite field, such an algorithm exists in the form of simple brute-force search.
- Possibly related: Vector space bases without axiom of choice