ScalingStacks

Lemma 1.6 . [025J]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Lemma 1.6.

Let WW be a subspace of VV and ψ∈W∨:=Homk​(W,k)\psi\in W^{\vee}:=\mathrm{Hom}_{k}(W,k). For any α∈(0,1)\alpha\in(0,1), there is φ∈V∨:=Homk​(V,k)\varphi\in V^{\vee}:=\mathrm{Hom}_{k}(V,k) such that φ|W=ψ\left.{\varphi}\right|_{{W}}=\psi and

‖ψ‖∨≤‖φ‖∨≤α−1​‖ψ‖∨.\|\psi\|^{\vee}\leq\|\varphi\|^{\vee}\leq\alpha^{-1}\|\psi\|^{\vee}.

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.