Definitions
1.NounA syntype or paratype.
2.NounA Banach space
(
X
,
‖
⋅
‖
)
{\displaystyle (X,\|\cdot \|)}
, such that, given a sequence of independent random variables from a Rademacher distribution,
(
ε
i
)
{\displaystyle (\varepsilon _{i})}
, and a rank
q
{\displaystyle q}
in the range from 2 to infinity, there exists a finite constant
C
≥
1
{\displaystyle C\geq 1}
such that for all finite sequences
(
x
i
)
∈
X
{\displaystyle (x_{i})\in X}
,
E
ε
[
‖
∑
i
=
1
n
ε
i
x
i
‖
q
]
≥
1
C
q
(
∑
i
=
1
n
‖
x
i
‖
q
)
,
if
2
≤
q
<
∞
{\displaystyle \mathbb {E} _{\varepsilon }\left[\left\|\sum \limits _{i=1}^{n}\varepsilon _{i}x_{i}\right\|^{q}\right]\geq {\frac {1}{C^{q}}}\left(\sum \limits _{i=1}^{n}\|x_{i}\|^{q}\right),\quad {\text{if}}\;2\leq q<\infty }
or
E
ε
[
‖
∑
i
=
1
n
ε
i
x
i
‖
]
≥
1
C
sup
‖
x
i
‖
,
if
q
=
∞
{\displaystyle \mathbb {E} _{\varepsilon }\left[\left\|\sum \limits _{i=1}^{n}\varepsilon _{i}x_{i}\right\|\right]\geq {\frac {1}{C}}\sup \|x_{i}\|,\quad {\text{if}}\;q=\infty }
.