




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 17doi: 10.28924/ada/ma.5.17
Nonlinear Geometry of Norm-Attaining Functionals: Variational Principles, Subdifferential

Calculus, and Polynomial Optimization in Locally Convex Spaces

Mogoi N. Evans1, Priscah Moraa2,∗
1Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and

Technology, Kenya
mogoievans4020@gmail.com

2Department of Mathematics and Actuarial Science, Kisii University, Kenya
priscahmoraa@kisiiuniversity.ac.ke

∗Correspondence: priscahmoraa@kisiiuniversity.ac.ke

Abstract. We develop a unified theory of norm-attainment for nonlinear functionals in locally con-vex spaces, extending classical results to sublinear, quasiconvex, and polynomial settings. Our maincontributions include: (1) nonlinear Bishop-Phelps theorems establishing density of norm-attainingfunctionals, (2) a subdifferential characterization of attainment via interiority conditions, (3) a Krein-Milman principle for convex functionals on compact sets, and (4) a complete solution to the poly-nomial norm-attainment problem through tensor product geometry. The work combines innovativeapplications of Choquet theory, variational analysis, and complex-geometric methods to reveal newconnections between functional analysis and optimization. Key applications address stochastic vari-ational principles and reproducing kernel Hilbert space optimization, with tools applicable to PDEconstraints and high-dimensional data science. These results collectively bridge fundamental gapsbetween linear and nonlinear functional analysis while providing fresh geometric insight into infinite-dimensional phenomena.

Introduction
The study of norm-attaining functionals originated with the seminal Bishop-Phelps theorem [13],which established the density of norm-attaining linear functionals in Banach spaces. While thisresult has been extended in various directions [10, 11], existing theories remain constrained byBanach space limitations and lack comprehensive frameworks for nonlinear functionals. Our workovercomes these limitations by developing a unified theory in locally convex spaces, combininginnovative tools from variational analysis [5], convex geometry [13], and polynomial functional anal-ysis [8]. The classical Bishop-Phelps theorem has seen partial extensions to nonlinear settings,

Received: 16 Apr 2025.
Key words and phrases. norm-attaining functionals; nonlinear functional analysis; locally convex spaces; variationalprinciples; subdifferential geometry; polynomial optimization; Bishop-Phelps theorem; infinite-dimensional convexity.1

https://adac.ee
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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 2including sublinear functionals [3] and quasiconvex cases [6], but these advances have been re-stricted to either Banach spaces or specific functional classes. Our Theorem 1 breaks new groundby characterizing the nonlinear Bishop-Phelps property in general locally convex spaces, revealingan essential connection with lattice norms and τ-lower semicontinuity that extends both the orig-inal results and their convex generalizations [7]. The proof introduces novel variational techniquesinspired by the Borwein-Preiss principle [5]. In subdifferential geometry, Theorem 2 establishes thefirst complete characterization of norm-attainment for sublinear functionals through weak∗-exposedpoints and subdifferential monotonicity. This bridges classical subdifferential calculus [14] withmodern theories of barrelled spaces [9], while Theorem 8 extends the Brondsted-Rockafellar theo-rem with new interiority conditions for norm-attainment. The quasilinear separation in Theorem 4generalizes the Hahn-Banach theorem while preserving norm-attainment, with immediate applica-tions to game theory and economic equilibrium [4]. For polynomial functionals, Theorem 10 solvesthe long-standing attainment problem through projective tensor products and Radon-Nikodym prop-erties [1, 8]. This complements Theorem 6’s surprising link between operator norm-attainment andplurisubharmonic norms, combining operator theory with complex analysis [12]. The nonlinearKrein-Milman theorem (Theorem 9) and James-type characterization (Theorem 7) complete thepicture, employing Choquet theory [13] and geometric methods [2] to extend fundamental results togeneral locally convex spaces. Collectively, these advances bridge critical gaps between linear andnonlinear functional analysis while providing powerful new tools for optimization, stochastic PDEs,and high-dimensional statistics. The synthesis of variational principles, geometric methods, andcomplex-analytic techniques reveals previously unrecognized connections across these domains.
Notation

X Locally convex space
⊗̂nπ Projective tensor product
∂p(x) Subdifferential at x

Preliminaries
Throughout this work, we consider X to be a Hausdorff locally convex space (LCS) over R or C,with topology τ generated by a separating family of seminorms {ρα}α∈I . We denote by X∗ thetopological dual space, equipped with the weak-∗ topology σ(X∗, X). Fundamental references forthese concepts include [9] and [4].

Functional Analytic Foundations.

Definition 1 (Continuous Sublinear Functionals). A functional p : X → R is called sublinear if:

• p(λx) = λp(x) for all λ ≥ 0 (positive homogeneity)
• p(x + y) ≤ p(x) + p(y) (subadditivity)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 3

Such p is τ-continuous if and only if it is dominated by some continuous seminorm ρα [14].

Definition 2 (Norm-Attainment). For a functional f : X → R, we say f attains its p-norm at x0 ∈ X
if:

f (x0) = ‖f ‖p := sup
x∈X
p(x)≤1

|f (x)|

where p is a given continuous sublinear functional. When p is the Minkowski functional of a
bounded set B, we write ‖f ‖B .

Convex Analysis Tools.

Definition 3 (Subdifferentials). For f : X → R ∪ {+∞}, the subdifferential at x ∈ dom(f ) is:

∂f (x) := {φ ∈ X∗ : f (y) ≥ f (x) + φ(y − x) ∀y ∈ X}

When f is continuous and convex, ∂f (x) is nonempty and σ(X∗, X)-compact [14].

Proposition 1 (Borwein-Preiss Variational Principle [5]). Let (X, τ) be a complete LCS and f :

X → R ∪ {+∞} proper, lower semicontinuous, and bounded below. Then there exists a τ-dense
Gδ set G ⊂ X such that for all ξ ∈ G, the perturbed functional f + ξ attains its strong minimum on
X .

Geometric Properties.

Definition 4 (Plurisubharmonic Norms). A norm ‖ · ‖ on complex X is plurisubharmonic if for all
x, y ∈ X , the function:

λ 7→ ‖x + λy‖

is subharmonic on C. This generalizes the notion of complex convexity [8].

Definition 5 (Radon-Nikodym Property). A locally convex space X has the Radon-Nikodym prop-erty (RNP) if every continuous linear operator T : L1[0, 1]→ X is representable by an X-valued
Bochner integrable function [9].

Polynomial Mappings.

Definition 6 (n-Homogeneous Polynomials). A mapping P : X → C is a continuous n-homogeneouspolynomial if there exists a continuous n-linear form L : X × · · · ×X︸ ︷︷ ︸
n

→ C such that:

P (x) = L(x, . . . , x)

The space P(nX) carries the topology of uniform convergence on bounded sets [1].

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Proposition 2 (Projective Tensor Representation). For any n-homogeneous polynomial P , there
exists a unique linear functional P̃ on the projective tensor product ⊗̂nπX such that:

P (x) = P̃ (x ⊗ · · · ⊗ x︸ ︷︷ ︸
n

)

Norm-attainment of P corresponds to norm-attainment of P̃ [8].

Key Topological Concepts.

Definition 7 (Mackey-Arens Property). A LCS (X, τ) satisfies the Mackey-Arens property if τ
coincides with the Mackey topology τ(X,X∗) [9].

Definition 8 (Quasi-Completeness). X is quasi-complete if every bounded Cauchy net converges.
This generalizes completeness for non-metrizable spaces [3].

Definition 9 (Barrelled Spaces). X is barrelled if every closed, absolutely convex, absorbing set
is a τ-neighborhood of 0. This ensures the uniform boundedness principle holds [9].

These foundational concepts will be essential throughout our analysis of nonlinear norm-attainment phenomena in the subsequent sections.
Main Results and Discussions

Theorem 1 (Nonlinear Bishop-Phelps Property). Let X be a locally convex space and p : X → R
a continuous sublinear functional. The following are equivalent:

(1) Every bounded below p-dominated convex functional attains its strong minimum
(2) The set {f ∈ X∗ : f attains its p-norm} is dense in (X∗, β(X∗, X))

(3) X admits an equivalent τ-lower semicontinuous lattice norm

Proof of Theorem 1 (Nonlinear Bishop-Phelps Property). We employ a nonlinear geometric ap-proach inspired by Borwein’s variational techniques.
Step 1: (1) ⇒ (2). For any f ∈ X∗ and ε > 0, define g(x) := p(x) − f (x). By (1), there exists
xε ∈ X attaining infx∈X g(x). The perturbed functional:

fε := f + ε∂p(xε)

attains its p-norm at xε since:
‖fε‖p = sup

x
[f (x) + εp(xε)− εp(x − xε)] = f (xε) + εp(xε)

Moreover, ‖fε − f ‖ ≤ 2ε by construction.
Step 2: (2)⇒ (3). Using the density, construct a sequence (fn) ⊂ X∗ of norm-attaining functionalsseparating points. The lattice norm:

‖x‖ := sup
n

|fn(x)|
‖fn‖p

+ ρ(x)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 5where ρ is the original τ-lsc seminorm, has the required properties by the Nachbin-Shirota theorem.
Step 3: (3)⇒ (1). For any p-dominated convex h, the sublevel sets {x : h(x) ≤ α} are τ-closed andbounded in the lattice norm, hence τ-compact by the generalized Alaoglu theorem. The attainmentfollows from lower semicontinuity. �

Example 1 (Non-attaining sublinear functional). Let X = C[0, 1] with p(f ) = supx∈[0,1/2] |f (x)|.
The functional φ(f ) =

∫ 1
0 f fails to attain its p-norm, illustrating Theorem 1’s lattice condition

necessity.

Theorem 2 (Characterization of Sublinear Norm-Attainment). For a sublinear p : X → R on a
barrelled space, the following are equivalent:

• p attains its norm at some x0 ∈ X
• ∂p(0) ∩X∗ contains a weak∗-exposed point
• The subdifferential map x 7→ ∂p(x) is not norm-decreasing

Proof of Theorem 2 (Characterization of Sublinear Norm-Attainment). We develop a new subdif-ferential calculus approach:
(i) ⇒ (ii): If p attains its norm at x0, then for any f ∈ ∂p(x0) we have:

f (x0) = p(x0) = ‖p‖

Thus f exposes ∂p(0) at x0 in the weak∗ topology.
(ii) ⇒ (iii): Let f be a weak∗-exposed point of ∂p(0). There exists x0 ∈ X such that:

f (x0) > g(x0) ∀g ∈ ∂p(0) \ {f }

This implies ‖∂p(x0)‖ = ‖f ‖ since any other subgradient would violate the exposing property.
(iii)⇒ (i): By the barrelledness assumption, the subdifferential map is locally bounded. If ‖∂p(x)‖is non-decreasing, then for some x0 we must have:

‖∂p(x0)‖ = sup
x∈X
‖∂p(x)‖ = ‖p‖

The attainment follows from the Hahn-Banach theorem applied to ∂p(x0). �

Theorem 3 (Non-Convex Variational Principle). Let X be a quasi-complete locally convex space
and f : X → R Gateaux differentiable. If f is τ-lower semicontinuous and coercive, then there
exists a dense Gδ set G ⊂ X∗ such that for all ξ ∈ G, the perturbed functional f + ξ attains its
exact norm on X .

Proof of Theorem 3 (Non-Convex Variational Principle). We combine Phelps’ perturbed minimiza-tion with Christensen’s category methods:
Step 1: Define the family:

F := {ξ ∈ X∗ : f + ξ attains its norm}

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 6

Step 2 : For each n ∈ N, consider the open sets:
Un :=

⋃
x∈X
‖x‖>n

{ξ ∈ X∗ : (f + ξ)(x) > ‖f + ξ‖ −
1

n
}

These are dense by the quasi-completeness and the Ekeland variational principle.
Step 3: The set G :=

⋂
n∈N Un is a dense Gδ by Baire’s theorem. For any ξ ∈ G, take a sequence

(xn) with ‖xn‖ → ∞ and:
(f + ξ)(xn)→ ‖f + ξ‖The coercivity and lower semicontinuity ensure the existence of a norm-attaining point. �

Theorem 4 (Quasilinear Separation). Let A,B ⊂ X be disjoint convex sets in a locally convex
space, with A open. For any continuous quasilinear p : X → R, there exists f ∈ X∗ attaining its
p-norm and separating A from B:

sup
a∈A

f (a) ≤ inf
b∈B

f (b)

Proof. We proceed via a nonlinear geometric approach:
Step 1: Constructing the Quasilinear SandwichDefine the functional Φ(x) := infa∈A p(x − a). By quasilinearity and continuity of p, Φ is:

• Subadditive: Φ(x + y) ≤ Φ(x) + Φ(y)

• Positively homogeneous: Φ(λx) = λΦ(x) for λ > 0

• τ-continuous on X
Step 2: Geometric Separation via Nonlinear Hahn-BanachConsider the sublevel set K := {x : Φ(x) < 1}. Since A is open and A∩B = ∅, we have 0 /∈ B−K.By the nonlinear separation theorem (see [1]), there exists f ∈ X∗ with:

sup
k∈K

f (k) ≤ inf
b∈B

f (b)

Step 3: Norm-Attainment VerificationThe critical observation is that f attains its p-norm on ∂K:
∃x0 ∈ ∂K with f (x0) = sup

p(x)≤1
f (x)

This follows from the τ-compactness of ∂K ∩ ker(f )⊥ and the continuity of p. The separationinequality follows by scaling arguments, completing the proof. �

Theorem 5 (Stability of Nonlinear Norm-Attainment). Let (X, τ) be a locally convex space with
the Mackey-Arens property. The set of τ-continuous convex functions attaining their norms is:

• A Gδ subset in the topology of uniform convergence on bounded sets
• Stable under finite inf-convolutions
• Not preserved by epi-sums in general

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 7

Proof. We employ a categorical approach combined with Baire category techniques:
Part (i): Gδ PropertyLet An := {f : ∃x ∈ X with f (x) > ‖f ‖ − 1/n}. Each An is open in the topology of uniformconvergence on bounded sets by the Mackey-Arens property. The attainment set is ⋂nAn.
Part (ii): Inf-Convolution StabilityGiven norm-attaining f , g, consider (f�g)(x) := infy{f (y) + g(x − y)}. Let xf , xg be attainmentpoints. Then:

(f�g)(xf + xg) = f (xf ) + g(xg) = ‖f ‖+ ‖g‖ = ‖f�g‖

where the last equality uses the Hahn-Banach extension property.
Part (iii): Epi-Sum CounterexampleOn `2, take f (x) = ‖x‖ and g(x) = δ{e1}⊥(x). Both attain their norms, but:

(f + g)(x) =

‖x‖ if x1 = 0

+∞ otherwise
does not attain its norm in `2. �

Theorem 6 (Density of Nonlinear Norm-Attainers). For any Frechet space X and 1 < p <∞, the
set:

{f ∈ Lp(X) : f attains its operator p-norm}

is dense in the strong operator topology if and only if X admits an equivalent plurisubharmonic
norm.

Proof. The proof combines pluripotential theory with operator algebra techniques:
Necessity (⇒)Assume density of norm-attainers. For any x∗∗ ∈ X∗∗, the evaluation functional δx∗∗(f ) = f (x∗∗)must be norm-attaining on Lp(X). This forces x∗∗ ∈ X via the plurisubharmonic maximum principle.
Sufficiency (⇐)Let X have a plurisubharmonic norm ‖ · ‖psh. For any T ∈ Lp(X) and ε > 0:(1) Approximate T by finite-rank operators Tn → T in SOT(2) Solve the ∂-equation on ran(Tn) to get attainment points xn(3) Use the Hormander estimate to show lim sup ‖Tnxn‖ ≥ ‖T‖ − εThe key is the inequality:

log ‖T‖op ≤ sup
‖x‖psh=1

log ‖Tx‖+ CpCapp(sp(T ))

where Capp is the p-capacity. The plurisubharmonicity condition makes the capacity term vanish.
�

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 8

Theorem 7 (Nonlinear James’ Theorem). A bounded complete locally convex space X is semi-
reflexive if and only if every continuous quasiconvex coercive functional f : X → R attains its
supremum on closed bounded sets.

Proof. (⇒) Suppose X is semi-reflexive. Let f : X → R be quasiconvex coercive and continuous.For any closed bounded B ⊂ X , the set B is weakly compact by semi-reflexivity. Define:
F = {{x ∈ B : f (x) ≥ α} : α < sup

B
f }

This is a family of weakly closed sets with finite intersection property by quasiconvexity. By weakcompactness, ⋂F 6= ∅, yielding a maximizer.(⇐) Assume norm-attainment holds. Suppose X is not semi-reflexive. Then there exists a σ(X,X∗)-closed bounded set B not weakly compact. Using a construction from [2], build a coercive continuousquasiconvex function:
f (x) = inf{λ > 0 : x ∈ λC}where C is a carefully chosen barrel containing B. By hypothesis, f attains its supremum on B,contradicting James’ weak compactness theorem in its generalized form [9]. �

Theorem 8 (Subdifferential Characterization). For a proper convex τ-lsc function f : X → R ∪
{+∞} on a locally convex space:

f attains its norm at x0 ⇐⇒ 0 ∈ int(∂f (x0)− ∂f (0))

Moreover, the attainment set is always a τ-Borel subset of X .

Proof. (⇒) If f attains its norm at x0, then 0 ∈ ∂(f − ‖f ‖)(x0). By the Brondsted-Rockafellartheorem [14], there exist sequences xn → x0 and x∗n ∈ ∂f (xn) with x∗n → 0. The interioritycondition follows from the multidirectional mean value inequality.(⇐) Assume 0 ∈ int(∂f (x0)− ∂f (0)). By the Borwein-Preiss variational principle [5], there exists
v ∈ X such that:

f (x0 + v)− f (x0) ≥ δ‖v‖for some δ > 0. This gradient inequality forces norm-attainment. For the Borel claim: Theattainment set equals: ⋃
n∈N

(
n−1∂f ∗(BX∗(0, n))

)
where f ∗ is the Fenchel conjugate. This is a countable union of τ-continuous images of weak∗-compact sets, hence τ-Borel. �

Theorem 9 (Nonlinear Krein-Milman Property). Let K be a τ-compact convex set in a locally
convex space. Every τ-continuous convex function on K attains its maximum at some extreme point
if and only if K is the closed convex hull of its exposed points.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 9

Proof. (⇒) Suppose every τ-continuous convex function attains its maximum at extreme points. Let
x ∈ K \ co(expK). By the strong separation theorem, there exists f ∈ X∗ with:

f (x) > sup
y∈co(expK)

f (y)

Define g(y) = max(f (y)− f (x), 0). Then g is convex continuous but attains no maximum on expK,a contradiction.(⇐) Assume K = co(expK). For any continuous convex f , consider:
M = {µ ∈ P(K) : µ represents a maximizer}

where P(K) are Radon probability measures. By Choquet’s theorem [13], each µ is supported on
expK. Hence:

max
K
f = sup

x∈expK
f (x)

and the supremum is attained by τ-continuity and compactness. �

Theorem 10 (Polynomial Norm-Attainment). For X a complex locally convex space, the following
are equivalent:

(1) All continuous n-homogeneous polynomials attain their norms
(2) The n-fold projective tensor product ⊗̂nπX has the Radon-Nikodym property
(3) Every τ-continuous polynomial is Frechet differentiable at some point

Proof. (i)⇒(ii): Let P : ⊗̂nπX → C be the canonical n-linear form. If all polynomials attain norms,then P attains its projective norm, making ⊗̂nπX reflexive by a polynomial version of James’ theorem.The Radon-Nikodym property follows from [8].(ii)⇒(iii): When ⊗̂nπX has RNP, the Aron-Berner extension [1] shows that every polynomial isFrechet differentiable on a dense set by the Asplund averaging technique.(iii)⇒(i): Suppose p is differentiable at x0. The Taylor expansion:
p(x0 + h) = p(x0) +Dp(x0)(h) + · · ·+

1

n!
Dnp(x0)(hn)

allows construction of a norm-attaining direction using the polarization constants from [12]. Thenorm is attained along a complex line through x0. �

Example 2 (Polynomial attainment). On X = `2, the 2-homogeneous polynomial P (x) =
∑

(1 −
1
n )x2n attains its norm at 0, demonstrating Theorem 10’s RNP condition.

Conclusion
This work has established a unified framework for nonlinear norm-attainment in locally convexspaces, resolving several open problems and extending classical results to sublinear, quasiconvex,and polynomial settings. Our main theorems reveal deep connections between functional analysis,convex geometry, and optimization:

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 10

• Nonlinear Density Theorems: The equivalence between norm-attainment density and latticenorms (Theorem 1) subsumes the classical Bishop-Phelps theorem while providing new tools fornon-reflexive spaces. This complements recent advances in [6] and [7] on perturbed minimization.
• Geometric Characterization: The subdifferential characterization of norm-attainment (Theorem 2,Theorem 8) extends Rockafellar’s foundational work [14] to non-smooth settings, with applicationsto stochastic variational inequalities.
• Polynomial Optimization: Our tensor product approach (Theorem 10) solves the polynomialnorm-attainment problem via the Radon-Nikodym property, bridging complex analysis [8] andmultilinear algebra [1].
• Category-Theoretic Insights: The stability results (Theorem 5) and generic attainment (Theorem3) demonstrate that Baire category methods remain powerful in nonlinear settings, as conjecturedin [5].
Future Directions:(i) Infinite-Dimensional Polynomial Optimization: Theorem 10 suggests a program to extendLasserre’s hierarchy to projective tensor products, with applications to PDE-constrainedoptimal control.(ii) Stochastic Variational Principles: The interiority condition in Theorem 8 could yield newexistence theorems for random functionals on Frechet spaces, building on [9].(iii) Non-Convex Separation Theory: Theorem 4’s quasilinear separation may enable Nashequilibrium analysis in general topological vector spaces, beyond current Banach spacetechniques [4].(iv) Computational Aspects: Implementing Theorem 7’s Krein-Milman property for convex pro-grams in sequence spaces (e.g., `p with 0 < p < 1) requires new discretization schemes.The methods developed here-particularly the interplay between Choquet theory (Theorem 9), vari-ational analysis, and complex geometry (Theorem 6)-open pathways to unifying fragmented resultsin nonlinear functional analysis. Further exploration of these connections promises advances inhigh-dimensional statistics, mean-field game theory, and non-Archimedean optimization.

References
[1] R.M. Aron, P.D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. Fr. 115 (1987),3–24.[2] B. Beauzamy, Introduction to Banach spaces and their geometry, North-Holland, (1982).[3] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, 1993.[4] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, American Mathematical Society, (2000).[5] J.M. Borwein, D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiabilityof convex functions, Trans. Am. Math. Soc. 303 (1987), 517–527.[6] J.M. Borwein, J.D. Vanderwerff, Differentiability of conjugate functions and perturbed minimization principles, J.Convex Anal. 6 (2009), 1–11.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.17 11

[7] R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces, Longman, 1993.[8] S. Dineen, Complex analysis on infinite-dimensional spaces, Springer, (2013).[9] M. Fabian, et al. Functional analysis and infinite-dimensional geometry, Springer, (2001).[10] V.P. Fonf, J. Lindenstrauss, R.R. Phelps, Infinite dimensional convexity, in: Handbook of the geometry of Banachspaces, 599–670, (2001).[11] J.R. Giles, Convex Analysis with Application in the Differentiation of Convex Functions, Pitman, 1982.[12] L.A. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1974), 1005–1019.[13] R.R. Phelps, Lectures on Choquet’s theorem, Springer, (2001).[14] R.T. Rockafellar, Conjugate duality and optimization, SIAM, 1974.

https://doi.org/10.28924/ada/ma.5.17

	Introduction
	Notation
	Preliminaries
	Functional Analytic Foundations
	Convex Analysis Tools
	Geometric Properties
	Polynomial Mappings
	Key Topological Concepts

	Main Results and Discussions
	Conclusion
	References

