On the existence of vectors dual to a set of linear functionals
aa r X i v : . [ m a t h . F A ] J u l ON THE EXISTENCE OF VECTORS DUAL TO A SET OF LINEAR FUNCTIONALS
SHIBO LIUA
BSTRACT . We give a simple proof of a crucial lemma that is established in [
1, Lemma2.1 ] by induction, and plays important roles in that paper and [ ] . In a recent paper [ ] , Brezhneva and Tret’yakov give an elementary proof of theKarush–Kuhn–Tucker Theorem (that is, [
1, Theorem 1.1 ] ) in normed linear spaces.This theorem is about the minimization problems with a finite number of inequalityconstraints. In their proof of [
1, Theorem 1.1 ] , an important step is to establish thefollowing lemma. Lemma 0.1 ( [
1, Lemma 2.1 ] ) . Let X be a linear space. Let ξ i : X → R , i =
1, . . . , n,be linear functionals which are linearly independent. Then there exists a set { η i } of nlinearly independent vectors in X such that the matrixA = 〈 ξ , η 〉 · · · 〈 ξ , η n 〉 ... ... 〈 ξ n , η 〉 · · · 〈 ξ n , η n 〉 is invertible. This lemma also plays a crucial role in [ ] , where an elementary proof of the La-grange multiplier theorem in normed linear spaces is given.We emphasize that 0.1 is equivalent to the following seemingly stronger result. Theorem 0.2.
Let X be a vector space. Let ξ i : X → R , i =
1, . . . , n, be linear functionalswhich are linearly independent. Then there exist ǫ , . . . , ǫ n in X such that 〈 ξ i , ǫ j 〉 = δ ij ,where δ ij is the Kronecker delta. In fact, let A − = b · · · b n ... b n · · · b nn , ǫ j = n X k = b kj η k Then AA − = I n , the n × n identity matrix, means 〈 ξ i , ǫ j 〉 = ® ξ i , n X k = b kj η k ¸ = n X k = 〈 ξ i , η k 〉 b kj = (cid:0) AA − (cid:1) ij = δ ij .Therefore, Lemma 0.1 and Theorem 0.2 are equivalent.In [ ] , Lemma 0.1 is proved by induction on n . In this note, we present a muchsimple proof of Theorem 0.2. Proof of Theorem 0.2.
We define a linear map ϕ : X → R n by ϕ ( x ) = (cid:0) ξ ( x ) , . . . , ξ n ( x ) (cid:1) . Mathematics Subject Classification.
Key words and phrases. linear functionals, linearly independent, linear map, perpendicular.This work was supported by NSFC (11671331). It was done while the author was visiting the AbdusSalam International Centre for Theoretical Physics. The author is grateful to ICTP for its hospitality.
We claim that ϕ is surjective. Otherwise, ϕ ( X ) is a proper subspace of R n , there exists λ ∈ R n \ { } which is perpendicular to ϕ ( X ) . Let λ = ( λ , . . . , λ n ) , then for all x ∈ X ,0 = λ · ϕ ( x ) = n X i = λ i ξ i ( x ) .Hence the set of linear functionals (cid:8) ξ i (cid:9) is linearly dependent, a contradiction.Let e j = (
0, . . . , 1, . . . , 0 ) be the j -th standard base vector of R n and take ǫ j ∈ ϕ − ( e j ) ,then (cid:0) ξ ( ǫ j ) , . . . , ξ n ( ǫ j ) (cid:1) = ϕ ( ǫ j ) = e j .It follows that ξ i ( ǫ j ) = δ ij . (cid:3) R EFERENCES [ ] O. Brezhneva, A. A. Tret’yakov, An elementary proof of the Karush-Kuhn-Tucker theorem in normed linear spaces for problems with a finite number of inequality constraints,Optimization 60 (2011) 613–618. [ ] O. Brezhneva, A. A. Tret’yakov, An elementary proof of the Lagrange multiplier theorem in normed linear spaces,Optimization 61 (2012) 1511–1517.S
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