# Observations on the Separable Quotient Problem for Banach Spaces

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## Abstract

**:**

**Proposition**

**1.**

- (i)
- X is weakly compactly generated.
- (ii)
- There is a Banach space Y, and a weak* to weak continuous linear injection $T:{Y}^{*}\to X$, with dense range.
- (iii)
- There is a Banach space Y, and a weak* to weak continuous linear injection $T:{X}^{*}\to Y$.
- (iv)
- There is a Banach space Y, and a weak* to weak continuous linear injection $T:{X}^{*}\to Y$, with dense range.
- (v)
- There is a Banach space Y, and a weak* to weak continuous linear operator $T:{Y}^{*}\to X$, with dense range.

**Proof.**

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**2.**

- (i)
- X has an infinite-dimensional separable quotient Banach space.
- (ii)
- X has a dense nonbarrelled subspace.
- (iii)
- X has a separable infinite-dimensional quasicomplemented subspace.
- (iv)
- X has a proper quasicomplemented subspace.

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**1.**

**Proof.**

**Corollary**

**2.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Morris, S.A.; Yost, D.T.
Observations on the Separable Quotient Problem for Banach Spaces. *Axioms* **2020**, *9*, 7.
https://doi.org/10.3390/axioms9010007

**AMA Style**

Morris SA, Yost DT.
Observations on the Separable Quotient Problem for Banach Spaces. *Axioms*. 2020; 9(1):7.
https://doi.org/10.3390/axioms9010007

**Chicago/Turabian Style**

Morris, Sidney A., and David T. Yost.
2020. "Observations on the Separable Quotient Problem for Banach Spaces" *Axioms* 9, no. 1: 7.
https://doi.org/10.3390/axioms9010007