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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:

Hyperplane
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  • There's two isomorphic vector spaces $(V, B)$ and $(V, B')$. Requiring that we don't know the representation of $B$ in $B'$ makes me think vector space operations won't really work, since the operations are different in $(V, B)$ than in $(V, B')$, and we are not given a way of relating them, like a linear transformation. I suppose in the finite dimensional case, one could try to optimize $(a_1, \ldots, a_n)$ such that $|\vec{v} - a_1 e_1 + \cdots + a_n e_n|$ is at a minimum. But, the finite dimensional case always has an inner product for $\mathbb{R}$ and $\mathbb{C}$. Just my thoughts. – J126 Feb 12 '22 at 23:09
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    @J126 The finite dimensional case need not have an inner product, e.g. if $K$ is a finite field. The idea of minimizing the residual $v-λeₐ$ is intuitive, however, $|\vec{v} - a_1 e_1 + \cdots + a_n e_n|$ is undefined without assuming a norm structure. But overall, I think the idea is right that without an inner product there seems to be no automatism for relating $B$ to $B'$, the question is how to prove it. – Hyperplane Feb 12 '22 at 23:17
  • My comment about finite dimensional vector spaces specified that it was over $\mathbb{R}$ or $\mathbb{C}$, where there is always an inner product. As above, if you are giving no data to relate $B$ to $B'$, how can one expect to compute a change in coefficients? – J126 Feb 13 '22 at 01:47
  • @J126 How do you construct inner product for a general vector space over $\mathbb{C}$ or $\mathbb{R}$? The only construction I am aware of uses a basis and has the same problem of finding coordinates with respect to the basis. – Korf Feb 13 '22 at 14:45

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