# Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

## 2. Conditions of the Unboundedness

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**1.**

**Proposition**

**3.**

- 1.
- The range $\mathfrak{P}\left(X\right)$ of X under $\mathfrak{P}$ is in ${\ell}_{\infty}.$
- 2.
- $\mathfrak{P}$ maps the ball $r{\mathcal{B}}_{X}$ into the ball $r{\mathcal{B}}_{{\ell}_{\infty}},$ $r\ge 0.$
- 3.
- If z is in the range of $\mathfrak{P}\left(x\right),$ then ${P}_{n}\left(x\right)={I}_{n}\left(z\right).$

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

- 1.
- If $\mathfrak{P}$ maps X onto ${\ell}_{\infty},$ then the algebra ${\mathcal{P}}_{\mathbb{P}}\left(X\right)$ does not support analytic functions of unbounded type.
- 2.
- If $\mathfrak{P}$ maps X to ${c}_{0},$ then$$\sum _{n=1}^{\infty}{P}_{n}\left(x\right)\in H\left(X\right)\backslash {H}_{b}\left(X\right).$$
- 3.
- If there is $x\in X$ such that $\mathfrak{P}\left(x\right)\subset {\ell}_{\infty}\backslash {c}_{0},$ then$$\sum _{n=1}^{\infty}{P}_{n}\left(x\right)\notin H\left(X\right).$$

**Proof.**

**Remark**

**1.**

**Example**

**1.**

**Example**

**2.**

**Corollary**

**1.**

**Proof.**

## 3. Lineability of ${\mathit{H}}_{\mathbb{P}}\left(\mathit{X}\right)\backslash {\mathit{H}}_{\mathit{b}\mathbb{P}}\left(\mathit{X}\right)$

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

- $\underset{k\to \infty}{lim}{\phi}_{k}\left(x\right)=0$ for every $x\in X,$
- $\parallel {\phi}_{k}\parallel =1,$$k\in \mathbb{N},$
- ${sup}_{k\in \mathbb{N}}\parallel {e}_{k}\parallel <\infty ,$
- ${\phi}_{k}\left({e}_{j}\right)={\delta}_{kj},$ where $k,j\in \mathbb{N}$ and ${\delta}_{kj}$ is the Kronecker delta.

**Theorem**

**4.**

**Proof.**

**Corollary**

**3.**

**Proof.**

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Zagorodnyuk, A.; Hihliuk, A.
Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability. *Axioms* **2021**, *10*, 150.
https://doi.org/10.3390/axioms10030150

**AMA Style**

Zagorodnyuk A, Hihliuk A.
Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability. *Axioms*. 2021; 10(3):150.
https://doi.org/10.3390/axioms10030150

**Chicago/Turabian Style**

Zagorodnyuk, Andriy, and Anna Hihliuk.
2021. "Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability" *Axioms* 10, no. 3: 150.
https://doi.org/10.3390/axioms10030150