MATHLIBANNEX / CANONICAL DECLARATION CARD

Compactness of the signed generator set

MathlibAnnex.PluckerBody.isCompact_generators

theorem

Passes compactness from contractions to their two signed minor images.

Statement

Let and give and their coordinate sup norms. Let be the fixed coordinate Lebesgue measure on . Let be a continuous real seminorm with specified constants such that

The lower bound makes a norm. Its closed unit ball and real reference volume are Here and below a finite Lebesgue measure is read as a real number when used as a scalar.

Let be the finite set of increasing -tuples of distinct rows from . The minor-coordinate space carries its sup norm. For a linear map , put in the standard coordinate basis. Write for its maximal-minor vector and for that vector scaled by . Their coordinates and scaling are

A -contraction is a linear map satisfying for every . Let be their set, and put

Then is compact.

Assumptions

The source and target are finite real coordinate spaces with sup norms. The seminorm is continuous and has the stated positive lower and upper comparison constants. The measure is the original coordinate Lebesgue measure. The dimensions may be zero. When , the unique empty minor has determinant ; when , the minor-index set is empty and its function space has one element. Neither case is excluded.

Conclusion

Compactness holds in the usual topology of the finite-coordinate space .

Proof route

Bound the operators and then take their continuous signed images.

Proof steps

  1. Make the contraction set compact. Give the finite-dimensional operator space its induced operator norm

    Then

    For fixed , evaluation is continuous, so every set in the intersection is closed. The norm bound follows from . Thus the contraction set is closed and contained in the closed operator ball of radius . Finite dimensionality makes it compact.

  2. Take two continuous images. Put . Then

    This is the image-union identity. Matrix entries are continuous in the operator, determinants are finite polynomials, and the scalar is fixed. These are continuity of maximal minors and continuity after volume scaling. The continuous-image theorem applied first to and then to negation on makes and compact. Substitution into the displayed union gives compactness of .

Main citations

Lean source signature (exact)

/-- The generator set is compact at every pair of finite dimensions. -/
theorem isCompact_generators {n N : ℕ} (M : EquivalentSeminorm (Fin n → ℝ)) :
    IsCompact (generators M N)

There is no separate hypothesis after M. The conclusion begins after the final colon.

In the source Mathematical meaning
{n N : ℕ} Arbitrary nonnegative integers . Braces mean that Lean may infer these parameters; they are not extra assumptions.
Fin n → ℝ; Fin N → ℝ The spaces and with their sup norms. A vector is a list of real coordinates, indexed from in Lean and from in the formulas here.
M : EquivalentSeminorm (Fin n → ℝ) The continuous seminorm , with , and for every .
Matrix.MaximalMinorIndex n (Fin N) The index set : choose distinct rows among the rows and read them in increasing order. A function of this type into is a minor-coordinate vector .
M.closedUnitBall; M.closedUnitBallVolume and , with the finite measure read as a real number.
Matrix.ballVolumeScaledMaximalMinors M B , whose coordinate is .
In the source Mathematical meaning
generators M N The set . Its elements are minor-coordinate vectors, not the maps .
IsCompact (generators M N) The set is compact in the finite-coordinate space with its usual topology.

Exact definitions used in this reading: Signed generator set.

Exact surrounding binder context (separate excerpts)

Exact source lines 12–14:

noncomputable section

namespace MathlibAnnex.PluckerBody

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