By Scott Messick

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**Example text**

First note that, if one of a0 or b0 is zero, let us say γ0 = a0 · e, then Sγ0 = {Z = β1 · e | |β1 | = a0 }, and this set is a circumference in the real two–dimensional plane BCe with center a0 at the origin and radius √ = |a0 · e|. Similarly, if γ0 = b0 · e† , the set Sγ0 is a 2 b0 circumference in BCe† with center at the origin and radius √ = |b0 · e† |. 2 16 Fig. 5 A Partial Order on D and a Hyperbolic-Valued Norm 17 If a0 = 0 and b0 = 0 then the intersection of the hyperbolic plane D and the sphere Sγ0 consists of exactly four hyperbolic numbers: ±a0 e ± b0 e† , where the plane D touches the sphere tangentially.

The next step is to prove what the analog of being “Hermitian for an inner product” is in our situation. y, x⊗→X = ey1 + e† y2 , ex1 + e† x2 ⊗→X = e y1 , x1 ⊗1 + e† y2 , x2 ⊗2 → = e y1 , x1 ⊗1 + e† y2 , x2 ⊗2 = e x1 , y1 ⊗1 + e† x2 , y2 ⊗2 = x, y⊗ X . Note that the “inner product square” 4 Norms and Inner Products on BC-Modules 44 x, x⊗ X = e∈x1 ∈21 + e† ∈x2 ∈22 ∗ D+ . 7) is always a positive hyperbolic number, without a need for any additional requirements on ·, ·⊗1 and ·, ·⊗2 . 7) always verifies.

But we are not aware of any work that treats the convergence with respect to a norm with hyperbolic values. 4 A sequence of bicomplex numbers {Z n }n∈N converges to the bicomplex number Z 0 with respect to the hyperbolic-valued norm | · |k if for all ε > 0 there exists N ∈ N such that for all n ≥ N there holds: |Z n − Z 0 |k ε. 12), it follows that a sequence {Z n }n∈N converges to the bicomplex number Z 0 with respect to the hyperbolic-valued norm if and only if it converges to Z 0 with respect to the Euclidean norm, and so even though the two norms cannot be compared as they take values in different rings, one still obtains the same notion of convergence.

### Limits in category theory by Scott Messick

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