Manuscript IX-C@1 · IX-C.tex · section · 03ce8e6b

Both admitted partial stages

Both admitted partial stages · lines 587–610

IX-C@1#eq:stages

Define actual-functional subspaces

L_{4} = span_{C} {B_{f} W_{g}}, L_{2} = span_{C} {Q_{f, g}},

R_{4} = span_{C} {W_{g} B_{f^{'}}}, R_{2} = span_{C} {Q^{R}_{g, f^{'}}},

and

L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2} .

The sums are direct because the two summands are homogeneous of field degrees four and two. The underlying pointwise spaces $L_{4}, R_{4}$ coincide when the test ranges coincide. We retain the ordered notation to distinguish their roles. Likewise, their common elements must not be assigned inconsistent values by any future product.

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IX-C@1#eq:stages · line 599
$L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2} .$
unlabeled · line 590
$L_{4} = span_{C} {B_{f} W_{g}}, L_{2} = span_{C} {Q_{f, g}},$
unlabeled · line 594
$R_{4} = span_{C} {W_{g} B_{f^{'}}}, R_{2} = span_{C} {Q^{R}_{g, f^{'}}},$

Both admitted partial stages

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