Exterior Norm Conjecture Table
We're trying to figure out how induced norms on the exterior algebra of a finite-dimensional Banach space work in this Facebook thread.
This paper defines two interesting norms on the exterior algebra which are variants of Grothendieck's norms on tensor algebras. Although the paper focuses on arbitrary Banach spaces, we'll limit our attention here to only finite-dimensional ones.
Given a norm on a finite-dimensional Banach space V, the injective norm is given as follows: if v<i> is the restriction of v to grade i, then the injective norm is
[math]\sup_i\{\sup \left \{ v_{\langle i \rangle}(a_1,a_2,...,a_n) \mid a_i \in B_{V^*} \right \}\}[/math]
The norm that's dual to the above norm is called the projective norm.
The paper linked above has shown that the injective norm is a subspace of Grothendieck's original injective norm, defined on the tensor algebra T(V) - specifically, it's its restriction to the subspace of alternating tensors, which is isomorphic to Λ(V) in a canonical way. Likewise, the projective norm is a quotient norm of Grothendieck's projective norm, specifically the quotient given by T(V)/((v⊗v)), where we're modding by the ideal of all elements generated by v⊗v.
The paper claims that the projective norm is guaranteed to give you a Banach algebra, meaning it is sub-multiplicative and obeys ||v∧w|| ≤ ||v||·||w||; this is not so with the injective norm.
In addition to the above, there's another special induced norm defined in the case where the original norm is the L2 norm; this is the Euclidean norm where the various wedge products of the orthonormal basis of V are also declared orthonormal. In this norm we simply take the L2 norm of all of the coefficients of the (possibly multi-graded) multivector. To generalize this to the case of an arbitrary Lp norm; I will declare the naive Lp norm of the exterior algebra of a Banach space to simply extend the Lp norm to the coefficients of the entire multivector in similar fashion.
We're trying to conjecture the formulas for these three norms and see if we can find a pattern. For now, we'll start with grade-2 multivectors in R^3. A background of yellow means that the formula is conjectured and not proven. Here's what we have for some 2-vector ||a b c>>:
Projective | Injective | Naive Lp | |
L1 | |a|+|b|+|c| | 2*max(|a|+|b|, |a|+|c|, |b|+|c|) | |a|+|b|+|c| |
L2 | sqrt(|a|^2 + |b|^2 + |c|^2) | sqrt(|a|^2 + |b|^2 + |c|^2) | sqrt(|a|^2 + |b|^2 + |c|^2) |
Linf | max(|a|,|b|,|c|) | max(|a|,|b|,|c|) |