Quotient of subspace theorem

Quotient of subspace theorem In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.[1] Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant. The induced norm || · || on E, defined by {displaystyle |e|=min _{yin e}|y|,quad ein E,} is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that {displaystyle {frac {sqrt {Q(e)}}{K}}leq |e|leq K{sqrt {Q(e)}}} for {displaystyle ein E,} with K > 1 a universal constant.

The statement is relative easy to prove by induction on the dimension of Z (even for Y=Z, X=0, c=1) with a K that depends only on N; the point of the theorem is that K is independent of N.

In fact, the constant c can be made arbitrarily close to 1, at the expense of the constant K becoming large. The original proof allowed {displaystyle c(K)approx 1-{text{const}}/log log K.} [2] Notes ^ The original proof appeared in Milman (1984). See also Pisier (1989). ^ See references for improved estimates. References Milman, V.D. (1984), "Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space", Israel seminar on geometrical aspects of functional analysis, Tel Aviv: Tel Aviv Univ., X Gordon, Y. (1988), "On Milman's inequality and random subspaces which escape through a mesh in Rn", Geometric aspects of functional analysis, Lecture Notes in Math., Berlin: Springer, 1317: 84–106, doi:10.1007/BFb0081737, ISBN 978-3-540-19353-1 Pisier, G. (1989), The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge: Cambridge University Press hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Banach spacesAsymptotic geometric analysisTheorems in functional analysis