On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

On the fair coin tossing process

Let Ω = {1, 0} and for each integer n ≥ 1 let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{= {(a}{}_1\text{, a}{}_2\text{, …, a}{}_{\mathrm{n}}\text{)|(a}{}_1\text{, a}{}_2\text{, … , a}{}_{\mathrm{n}}\text{) ? Ω}{}_{\mathrm{n}}\text{and}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{= k}}$ for all k = 0,1,…,n. Let {Y_m}_m≥1 be a sequence of i.i.d. random variables such that $\text{P(Y}{}_1\text{= 0) = P(Y}{}_1\text{= 1) =}\text{1}\text{2}$. For each A in $\text{Ω}{}_{\mathrm{n}}$, let T_A be the first occurrence time of A with respect to the stochastic process {Y_m}_m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in $\text{Ω}{}_{\mathrm{n}}$, there is an element B in $\text{Ω}{}_{\mathrm{n}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element B also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a₁, a₂,…,a_n) in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element C also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_A is less than T_C is greater than $\text{1}\text{2}$ if n ≠ 2m or n = 2m but a_i = a_{i + 1} for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process.     

Families of finite sets in which no set is covered by the union of two others

Let f^k(n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f^{2t ? 1}(n) ? f^2t(n + 1) ? $\text{(}{}_{\mathrm{t}}{}^{\mathrm{n}}\text{)}\text{(}{}_{\mathrm{t}}{}^{\mathrm{2t?1}}\text{)}$ with equalities holding iff there exists a Steiner-system S(t, 2t ? 1, n). The bounds are approximately best possile for k ? 6 and of correct order of magnitude for k >/ 7, as well, even if the corresponding Steiner-systems do not exist.Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i.e., the nonuniform case).     

On the length of optimal TSP circuits in sets of bounded diameter

Let V be a set of n points in R^k. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) l^k(n) are the extremal values of l(V) defined by l^k(n)=max{l(V)|V∈V_n^k}, where V_n^k={V|V?R^k,|V|=n, d(V)=1}. A set V∈V_n^k is “longest” if l(V)=l^k(n). In this paper, first some geometrical properties of longest sets in R² are studied which are used to obtain l²(n) for small n′s, and then asymptotic bounds on l^k(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δ^k(n)=max{δ(V)|V∈V_n^k}. It is easily observed that $\text{δ}{}^{\mathrm{k}}\text{(n)=O(n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{)}$. Hence, $\text{c}{}_{\mathrm{k}}\text{=}\text{lim sup}{}_{\mathrm{n\to \infty }}\text{δ}{}^{\mathrm{k}}\text{(n)n}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}$ exists. It is shown that for all $\text{n, c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{≤δ}{}^{\mathrm{k}}\text{(n)}$, and hence, for all $\text{n, l}{}^{\mathrm{k}}\text{(n)≥ c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>1?1</mtext><mtext>k</mtext>}}$. For k=2, this implies that $\text{l}{}^2\text{(n)≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which generalizes an observation of Fejes-Toth that $\text{lim}{}_{\mathrm{n\to \infty }}\text{l}{}^2\text{(n)n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. It is also shown that $\text{l}{}^{\mathrm{k}}\text{(n) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]nδ}{}^{\mathrm{k}}\text{(n) + o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{+ o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{)}$. The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that $\text{l}{}^2\text{(n)≤(}\text{πn}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ O(1)}$.     

On majorization and Schur products

Suppose A, D₁,…,D_m are n × n matrices where A is self-adjoint, and let $\text{X = Σ}{}^{\mathrm{m}}{}_{\mathrm{k\; =\; 1}}\text{D}{}_{\mathrm{k}}\text{AD}{}^1{}_{\mathrm{k}}$. It is shown that if $\text{ΣD}{}_{\mathrm{k}}\text{D}{}^1{}_{\mathrm{k}}\text{= ΣD}{}^1{}_{\mathrm{k}}\text{D}{}_{\mathrm{k}}\text{= I}$, then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D₁,…,D_m, a result is obtained in terms of weak majorization. If each D_k is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if b_ii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

Powers of matrices and idempotency

In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) $\text{(A+A}{}^1\text{)}\text{2}$ is idempotent of rank r, and tr_rA (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) A^s=A^t for some positive integers s≠t, and trA=rankA iff A is idempotent. (c) $\text{A(A1A)}{}^{\mathrm{s}}\text{= A(AA}{}^1\text{)}{}^{\mathrm{t}}$ for some integers s≠t iff $\text{AA}{}^1\text{=A}{}^1\text{A}$ is idempotent, while $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{= A(AA}{}^1\text{)}{}^{\mathrm{s}}$ for some integers s≠0 iff $\text{AA}{}^1\text{=A}{}^1\text{A}$. (d) $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{=A}{}^1\text{(AA}{}^1\text{)}{}^{\mathrm{t}}$ for some integers s≠t and rankA=trA iff A is Hermitian idempotent, while $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{= A}{}^1\text{(AA}{}^1\text{)}{}^{\mathrm{s}}$ for some integer s iff A is Hermitian. Here $\text{A}{}^1$ indicates the conjugate transpose of A, and P^-α is defined iff (P⁺)^α=(P^α)⁺ for all positive integers α and P⁺ is the Moore-Penrose inverse of P.     

On a function due to S. Chowla

Let θ(k, pⁿ) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ ? + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0}$ (mod pⁿ) has a nontrivial solution. It is shown that if k is sufficiently large and divisible by p but not by p ? 1, then $\text{θ(k, p}{}^{\mathrm{n}}\text{) ≤ k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We also obtain the average order of θ(k), the least s such that the above congruence has a nontrivial solution for every prime p and every positive integer n.     

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

The permanent function is used to determine geometrical properties of the set $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then $\text{F}$ corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of $\text{F}$. If A is fully indecomposable, then the dimension of $\text{F}$ equals σ(A) ? 2n + 1, where σ(A) is the number of 1's in A. The only two-dimensional faces of $\text{Ω}{}_{\mathrm{n}}$ are triangles and rectangles. For n ? 6, $\text{Ω}{}_{\mathrm{n}}$ has four types of three-dimensional faces. The facets of the faces of $\text{Ω}{}_{\mathrm{n}}$ are characterized. Faces of $\text{Ω}{}_{\mathrm{n}}$ which are simplices are determined. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is two-neighborly but not a simplex, then $\text{F}$ has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2^k. All k-dimensional faces with at least 2^k?1 + 1 vertices are determined.     

Spectral theory of commuting self-adjoint partial differential operators

Let Ω denote a connected and open subset of $\text{R}$ⁿ. The existence of n commuting self-adjoint operators H₁,…, H_n on $\text{L}{}^2\text{(Ω)}$ such that each H_j is an extension of $\text{−}\text{i∂}\text{∂x}{}_{\mathrm{j}}$ (acting on $\text{C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(Ω))}$ is shown to be equivalent to the existence of a measure μ on $\text{R}$ⁿ such that f → $\text{}\text{̂}$tf (the Fourier transform of f) is unitary from $\text{L}{}^2\text{(Ω)}$ onto Ω. It is shown that the support of μ can be chosen as a subgroup of $\text{R}$ⁿ iff H₁,…, H_n can be chosen such that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$. This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of $\text{R}$ⁿ by some subgroup, i.e., iff Ω is essentially a fundamental domain.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

A generalization of Lavoie's inequality concerning the sum of idempotent matrices

Let a complex pxn matrix A be partitioned as A′=(A′₁,A′₂,…,A′_k). Denote by ?(A), A′, and A^? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A⁽¹⁴⁾ be an inner inverse of A such that A⁽¹⁴⁾A is a Hermitian matrix. Let B=(A⁽¹⁴⁾₁,A⁽¹⁴⁾₂,…,A_k⁽¹⁴⁾) and $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$.Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A⁽¹⁴⁾ or $\text{A}{}_{\mathrm{i>}}\text{A}{}_{\mathrm{<mtext>j</mtext>}}{}^1\text{=0}$ for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$ is no longer true.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.