# Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem

^{1}

^{2}

## Abstract

**:**

## 1. Introduction and Preliminaries

- (i)
- $F\left(\lambda x\right)=\lambda F\left(x\right)$ for all $\lambda \in \mathbb{R}$ and $x\in X$,
- (ii)
- there exists a $\delta >0$ such that for any finite sequence ${\left({x}_{i}\right)}_{i=1}^{m}\subset X$, $m\in \mathbb{N}$ and $\lambda \in {\mathbb{R}}^{m}$,$${\u2225\sum _{i=1}^{m}{\lambda}_{i}F\left({x}_{i}\right)-F\left(\sum _{i=1}^{m}{\lambda}_{i}{x}_{i}\right)\u2225}_{Y}\le \delta \sum _{i=1}^{m}\left|{\lambda}_{i}\right|{\u2225{x}_{i}\u2225}_{X}.$$

## 2. Finite-Dimensional Twisted Sums

**Lemma**

**1.**

**Proof.**

## 3. Main Result

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Theorem**

**2**

**.**Let E be a Banach space and S a closed subspace of E. Let ${T}_{2}\left(E\right)$ be either the Gaussian or Rademacher type 2 constant of E and ${C}_{2}\left(S\right)$ either the Gaussian or Rademacher cotype 2 constant of S. Then, there exists a projection $P:E\to S$ with

## 4. Applications

**Theorem**

**3**

**.**Let $f:\mathbb{P}\left({\mathbb{C}}^{d}\right)\to \mathbb{P}\left({\mathbb{C}}^{d}\right)$ be a function that satisfies

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**1**

**.**Let $f:{B}_{{S}_{2}^{d}}\to {S}_{2}^{d}$ be a continuous function that satisfies

**Proof of Proposition**

**1.**

**Lemma**

**4.**

**Proof of Lemma**

**4.**

## 5. Discussion and Perspectives

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Upper bounds for the Rademacher type and cotype constants of the spaces ${l}_{r}^{d}$ and ${S}_{r}^{d}$. The Gaussian type and cotype for these spaces behave in the same way, up to a factor of $\sqrt{2/\pi}$, as the Rademacher type and cotype. For a Hilbert space, the type and cotype constants are always equal to one.

Type $\mathit{p}\in [1,2]$ | Cotype $\mathit{q}\in [2,\infty ]$ | |
---|---|---|

${l}_{1}^{d}$ | ${d}^{1-\frac{1}{p}}$ | $\sqrt{2}$ |

Hilbert space | 1 | 1 |

${l}_{\infty}^{d}$ | ${(4logd)}^{1-1/p}$ | ${d}^{1/q}$ |

${S}_{1}^{d}$ | ${d}^{1-1/p}$ | $\sqrt{e}$ |

${S}_{\infty}^{d}$ | ${(4logd)}^{1-1/p}$ | ${d}^{1/q}$ |

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**MDPI and ACS Style**

Cuesta, J.
Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem. *Symmetry* **2019**, *11*, 1107.
https://doi.org/10.3390/sym11091107

**AMA Style**

Cuesta J.
Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem. *Symmetry*. 2019; 11(9):1107.
https://doi.org/10.3390/sym11091107

**Chicago/Turabian Style**

Cuesta, Javier.
2019. "Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem" *Symmetry* 11, no. 9: 1107.
https://doi.org/10.3390/sym11091107