In mathematics , especially functional analysis , a Fréchet algebra , named after Maurice René Fréchet , is an associative algebra
A
{\displaystyle A}
over the real or complex numbers that at the same time is also a (locally convex ) Fréchet space . The multiplication operation
(
a
,
b
)
↦
a
∗
b
{\displaystyle (a,b)\mapsto a*b}
for
a
,
b
∈
A
{\displaystyle a,b\in A}
is required to be jointly continuous .
If
{
‖
⋅
‖
n
}
n
=
0
∞
{\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }
is an increasing family[a] of seminorms for
the topology of
A
{\displaystyle A}
, the joint continuity of multiplication is equivalent to there being a constant
C
n
>
0
{\displaystyle C_{n}>0}
and integer
m
≥
n
{\displaystyle m\geq n}
for each
n
{\displaystyle n}
such that
‖
a
b
‖
n
≤
C
n
‖
a
‖
m
‖
b
‖
m
{\displaystyle \left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}
for all
a
,
b
∈
A
{\displaystyle a,b\in A}
.[b] Fréchet algebras are also called B 0 -algebras.
A Fréchet algebra is
m
{\displaystyle m}
-convex if there exists such a family of semi-norms for which
m
=
n
{\displaystyle m=n}
. In that case, by rescaling the seminorms, we may also take
C
n
=
1
{\displaystyle C_{n}=1}
for each
n
{\displaystyle n}
and the seminorms are said to be submultiplicative :
‖
a
b
‖
n
≤
‖
a
‖
n
‖
b
‖
n
{\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}
for all
a
,
b
∈
A
.
{\displaystyle a,b\in A.}
[c]
m
{\displaystyle m}
-convex Fréchet algebras may also be called Fréchet algebras.[2]
A Fréchet algebra may or may not have an identity element
1
A
{\displaystyle 1_{A}
. If
A
{\displaystyle A}
is unital , we do not require that
‖
1
A
‖
n
=
1
,
{\displaystyle \|1_{A}\|_{n}=1,}
as is often done for Banach algebras .
Properties
Continuity of multiplication. Multiplication is separately continuous if
a
k
b
→
a
b
{\displaystyle a_{k}b\to ab}
and
b
a
k
→
b
a
{\displaystyle ba_{k}\to ba}
for every
a
,
b
∈
A
{\displaystyle a,b\in A}
and sequence
a
k
→
a
{\displaystyle a_{k}\to a}
converging in the Fréchet topology of
A
{\displaystyle A}
. Multiplication is jointly continuous if
a
k
→
a
{\displaystyle a_{k}\to a}
and
b
k
→
b
{\displaystyle b_{k}\to b}
imply
a
k
b
k
→
a
b
{\displaystyle a_{k}b_{k}\to ab}
. Joint continuity of multiplication is part of the definition of a Fréchet algebra. For a Fréchet space with an algebra structure, if the multiplication is separately continuous, then it is automatically jointly continuous.[3]
Group of invertible elements. If
i
n
v
A
{\displaystyle invA}
is the set of invertible elements of
A
{\displaystyle A}
, then the inverse map
{
i
n
v
A
→
i
n
v
A
u
↦
u
−
1
{\displaystyle {\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}
is continuous if and only if
i
n
v
A
{\displaystyle invA}
is a
G
δ
{\displaystyle G_{\delta }
set . Unlike for Banach algebras ,
i
n
v
A
{\displaystyle invA}
may not be an open set . If
i
n
v
A
{\displaystyle invA}
is open, then
A
{\displaystyle A}
is called a
Q
{\displaystyle Q}
-algebra . (If
A
{\displaystyle A}
happens to be non-unital , then we may adjoin a unit to
A
{\displaystyle A}
[d] and work with
i
n
v
A
+
{\displaystyle invA^{+}
, or the set of quasi invertibles[e] may take the place of
i
n
v
A
{\displaystyle invA}
.)
Conditions for
m
{\displaystyle m}
-convexity. A Fréchet algebra is
m
{\displaystyle m}
-convex if and only if for every , if and only if for one , increasing family
{
‖
⋅
‖
n
}
n
=
0
∞
{\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }
of seminorms which topologize
A
{\displaystyle A}
, for each
m
∈
N
{\displaystyle m\in \mathbb {N} }
there exists
p
≥
m
{\displaystyle p\geq m}
and
C
m
>
0
{\displaystyle C_{m}>0}
such that
‖
a
1
a
2
⋯
a
n
‖
m
≤
C
m
n
‖
a
1
‖
p
‖
a
2
‖
p
⋯
‖
a
n
‖
p
,
{\displaystyle \|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},}
for all
a
1
,
a
2
,
…
,
a
n
∈
A
{\displaystyle a_{1},a_{2},\dots ,a_{n}\in A}
and
n
∈
N
{\displaystyle n\in \mathbb {N} }
. A commutative Fréchet
Q
{\displaystyle Q}
-algebra is
m
{\displaystyle m}
-convex, but there exist examples of non-commutative Fréchet
Q
{\displaystyle Q}
-algebras which are not
m
{\displaystyle m}
-convex.
Properties of
m
{\displaystyle m}
-convex Fréchet algebras. A Fréchet algebra is
m
{\displaystyle m}
-convex if and only if it is a countable projective limit of Banach algebras. An element of
A
{\displaystyle A}
is invertible if and only if its image in each Banach algebra of the projective limit is invertible.[f] [10]
Examples
Zero multiplication. If
E
{\displaystyle E}
is any Fréchet space, we can make a Fréchet algebra structure by setting
e
∗
f
=
0
{\displaystyle e*f=0}
for all
e
,
f
∈
E
{\displaystyle e,f\in E}
.
Smooth functions on the circle. Let
S
1
{\displaystyle S^{1}
be the 1-sphere . This is a 1-dimensional compact differentiable manifold , with no boundary . Let
A
=
C
∞
(
S
1
)
{\displaystyle A=C^{\infty }(S^{1})}
be the set of infinitely differentiable complex-valued functions on
S
1
{\displaystyle S^{1}
. This is clearly an algebra over the complex numbers, for pointwise multiplication. (Use the product rule for differentiation .) It is commutative, and the constant function
1
{\displaystyle 1}
acts as an identity. Define a countable set of seminorms on
A
{\displaystyle A}
by
‖
φ
‖
n
=
‖
φ
(
n
)
‖
∞
,
φ
∈
A
,
{\displaystyle \left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,}
where
‖
φ
(
n
)
‖
∞
=
sup
x
∈
S
1
|
φ
(
n
)
(
x
)
|
{\displaystyle \left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}\left|\varphi ^{(n)}(x)\right|}
denotes the supremum of the absolute value of the
n
{\displaystyle n}
th derivative
φ
(
n
)
{\displaystyle \varphi ^{(n)}
.[g] Then, by the product rule for differentiation, we have
‖
φ
ψ
‖
n
=
‖
∑
i
=
0
n
(
n
i
)
φ
(
i
)
ψ
(
n
−
i
)
‖
∞
≤
∑
i
=
0
n
(
n
i
)
‖
φ
‖
i
‖
ψ
‖
n
−
i
≤
∑
i
=
0
n
(
n
i
)
‖
φ
‖
n
′
‖
ψ
‖
n
′
=
2
n
‖
φ
‖
n
′
‖
ψ
‖
n
′
,
{\displaystyle {\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}
where
(
n
i
)
=
n
!
i
!
(
n
−
i
)
!
,
{\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!},}
denotes the binomial coefficient and
‖
⋅
‖
n
′
=
max
k
≤
n
‖
⋅
‖
k
.
{\displaystyle \|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.}
The primed seminorms are submultiplicative after re-scaling by
C
n
=
2
n
{\displaystyle C_{n}=2^{n}
.
Sequences on
N
{\displaystyle \mathbb {N} }
. Let
C
N
{\displaystyle \mathbb {C} ^{\mathbb {N} }
be the space of complex-valued sequences on the natural numbers
N
{\displaystyle \mathbb {N} }
. Define an increasing family of seminorms on
C
N
{\displaystyle \mathbb {C} ^{\mathbb {N} }
by
‖
φ
‖
n
=
max
k
≤
n
|
φ
(
k
)
|
.
{\displaystyle \|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.}
With pointwise multiplication,
C
N
{\displaystyle \mathbb {C} ^{\mathbb {N} }
is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative
‖
φ
ψ
‖
n
≤
‖
φ
‖
n
‖
ψ
‖
n
{\displaystyle \|\varphi \psi \|_{n}\leq \|\varphi \|_{n}\|\psi \|_{n}
for
φ
,
ψ
∈
A
{\displaystyle \varphi ,\psi \in A}
. This
m
{\displaystyle m}
-convex Fréchet algebra is unital, since the constant sequence
1
(
k
)
=
1
,
k
∈
N
{\displaystyle 1(k)=1,k\in \mathbb {N} }
is in
A
{\displaystyle A}
.
Equipped with the topology of uniform convergence on compact sets , and pointwise multiplication,
C
(
C
)
{\displaystyle C(\mathbb {C} )}
, the algebra of all continuous functions on the complex plane
C
{\displaystyle \mathbb {C} }
, or to the algebra
H
o
l
(
C
)
{\displaystyle \mathrm {Hol} (\mathbb {C} )}
of holomorphic functions on
C
{\displaystyle \mathbb {C} }
.
Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let
G
{\displaystyle G}
be a finitely generated group , with the discrete topology . This means that there exists a set of finitely many elements
U
=
{
g
1
,
…
,
g
n
}
⊆
G
{\displaystyle U=\{g_{1},\dots ,g_{n}\}\subseteq G}
such that:
⋃
n
=
0
∞
U
n
=
G
.
{\displaystyle \bigcup _{n=0}^{\infty }U^{n}=G.}
Without loss of generality, we may also assume that the identity element
e
{\displaystyle e}
of
G
{\displaystyle G}
is contained in
U
{\displaystyle U}
. Define a function
ℓ
:
G
→
[
0
,
∞
)
{\displaystyle \ell :G\to [0,\infty )}
by
ℓ
(
g
)
=
min
{
n
∣
g
∈
U
n
}
.
{\displaystyle \ell (g)=\min\{n\mid g\in U^{n}\}.}
Then
ℓ
(
g
h
)
≤
ℓ
(
g
)
+
ℓ
(
h
)
{\displaystyle \ell (gh)\leq \ell (g)+\ell (h)}
, and
ℓ
(
e
)
=
0
{\displaystyle \ell (e)=0}
, since we define
U
0
=
{
e
}
{\displaystyle U^{0}=\{e\}
.[h] Let
A
{\displaystyle A}
be the
C
{\displaystyle \mathbb {C} }
-vector space
S
(
G
)
=
{
φ
:
G
→
C
|
‖
φ
‖
d
<
∞
,
d
=
0
,
1
,
2
,
…
}
,
{\displaystyle S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \},}
where the seminorms
‖
⋅
‖
d
{\displaystyle \|\cdot \|_{d}
are defined by
‖
φ
‖
d
=
‖
ℓ
d
φ
‖
1
=
∑
g
∈
G
ℓ
(
g
)
d
|
φ
(
g
)
|
.
{\displaystyle \|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.}
[i]
A
{\displaystyle A}
is an
m
{\displaystyle m}
-convex Fréchet algebra for the convolution multiplication
φ
∗
ψ
(
g
)
=
∑
h
∈
G
φ
(
h
)
ψ
(
h
−
1
g
)
,
{\displaystyle \varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),}
[j]
A
{\displaystyle A}
is unital because
G
{\displaystyle G}
is discrete, and
A
{\displaystyle A}
is commutative if and only if
G
{\displaystyle G}
is Abelian .
Non
m
{\displaystyle m}
-convex Fréchet algebras. The Aren's algebra
A
=
L
ω
[
0
,
1
]
=
⋂
p
≥
1
L
p
[
0
,
1
]
{\displaystyle A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]}
is an example of a commutative non-
m
{\displaystyle m}
-convex Fréchet algebra with discontinuous inversion. The topology is given by
L
p
{\displaystyle L^{p}
norms
‖
f
‖
p
=
(
∫
0
1
|
f
(
t
)
|
p
d
t
)
1
/
p
,
f
∈
A
,
{\displaystyle \|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,}
and multiplication is given by convolution of functions with respect to Lebesgue measure on
[
0
,
1
]
{\displaystyle [0,1]}
.
Generalizations
We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet space or an F-space .
If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC). A complete LMC algebra is called an Arens-Michael algebra.
Open problems
Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an
m
{\displaystyle m}
-convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture.[16]
Notes
^ An increasing family means that for each
a
∈
A
,
{\displaystyle a\in A,}
‖
a
‖
0
≤
‖
a
‖
1
≤
⋯
≤
‖
a
‖
n
≤
⋯
{\displaystyle \|a\|_{0}\leq \|a\|_{1}\leq \cdots \leq \|a\|_{n}\leq \cdots }
.
^ Joint continuity of multiplication means that for every absolutely convex neighborhood
V
{\displaystyle V}
of zero, there is an absolutely convex neighborhood
U
{\displaystyle U}
of zero for which
U
2
⊆
V
,
{\displaystyle U^{2}\subseteq V,}
from which the seminorm inequality follows. Conversely,
‖
a
k
b
k
−
a
b
‖
n
=
‖
a
k
b
k
−
a
b
k
+
a
b
k
−
a
b
‖
n
≤
‖
a
k
b
k
−
a
b
k
‖
n
+
‖
a
b
k
−
a
b
‖
n
≤
C
n
(
‖
a
k
−
a
‖
m
‖
b
k
‖
m
+
‖
a
‖
m
‖
b
k
−
b
‖
m
)
≤
C
n
(
‖
a
k
−
a
‖
m
‖
b
‖
m
+
‖
a
k
−
a
‖
m
‖
b
k
−
b
‖
m
+
‖
a
‖
m
‖
b
k
−
b
‖
m
)
.
{\displaystyle {\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}
^ In other words, an
m
{\displaystyle m}
-convex Fréchet algebra is a topological algebra , in which the topology is given by a countable family of submultiplicative seminorms:
p
(
f
g
)
≤
p
(
f
)
p
(
g
)
,
{\displaystyle p(fg)\leq p(f)p(g),}
and the algebra is complete.
^ If
A
{\displaystyle A}
is an algebra over a field
k
{\displaystyle k}
, the unitization
A
+
{\displaystyle A^{+}
of
A
{\displaystyle A}
is the direct sum
A
⊕
k
1
{\displaystyle A\oplus k1}
, with multiplication defined as
(
a
+
μ
1
)
(
b
+
λ
1
)
=
a
b
+
μ
b
+
λ
a
+
μ
λ
1.
{\displaystyle (a+\mu 1)(b+\lambda 1)=ab+\mu b+\lambda a+\mu \lambda 1.}
^ If
a
∈
A
{\displaystyle a\in A}
, then
b
∈
A
{\displaystyle b\in A}
is a quasi-inverse for
a
{\displaystyle a}
if
a
+
b
−
a
b
=
0
{\displaystyle a+b-ab=0}
.
^ If
A
{\displaystyle A}
is non-unital, replace invertible with quasi-invertible.
^ To see the completeness, let
φ
k
{\displaystyle \varphi _{k}
be a Cauchy sequence. Then each derivative
φ
k
(
l
)
{\displaystyle \varphi _{k}^{(l)}
is a Cauchy sequence in the sup norm on
S
1
{\displaystyle S^{1}
, and hence converges uniformly to a continuous function
ψ
l
{\displaystyle \psi _{l}
on
S
1
{\displaystyle S^{1}
. It suffices to check that
ψ
l
{\displaystyle \psi _{l}
is the
l
{\displaystyle l}
th derivative of
ψ
0
{\displaystyle \psi _{0}
. But, using the fundamental theorem of calculus , and taking the limit inside the integral (using uniform convergence ), we have
ψ
l
(
x
)
−
ψ
l
(
x
0
)
=
lim
k
→
∞
(
φ
k
(
l
)
(
x
)
−
φ
k
(
l
)
(
x
0
)
)
=
lim
k
→
∞
∫
x
0
x
φ
k
(
l
+
1
)
(
t
)
d
t
=
∫
x
0
x
ψ
l
+
1
(
t
)
d
t
.
{\displaystyle {\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}^{x}\psi _{l+1}(t)dt.\end{aligned}
^
We can replace the generating set
U
{\displaystyle U}
with
U
∪
U
−
1
{\displaystyle U\cup U^{-1}
, so that
U
=
U
−
1
{\displaystyle U=U^{-1}
. Then
ℓ
{\displaystyle \ell }
satisfies the additional property
ℓ
(
g
−
1
)
=
ℓ
(
g
)
{\displaystyle \ell (g^{-1})=\ell (g)}
, and is a length function on
G
{\displaystyle G}
.
^
To see that
A
{\displaystyle A}
is Fréchet space, let
φ
n
{\displaystyle \varphi _{n}
be a Cauchy sequence. Then for each
g
∈
G
{\displaystyle g\in G}
,
φ
n
(
g
)
{\displaystyle \varphi _{n}(g)}
is a Cauchy sequence in
C
{\displaystyle \mathbb {C} }
. Define
φ
(
g
)
{\displaystyle \varphi (g)}
to be the limit. Then
∑
g
∈
S
ℓ
(
g
)
d
|
φ
n
(
g
)
−
φ
(
g
)
|
≤
∑
g
∈
S
ℓ
(
g
)
d
|
φ
n
(
g
)
−
φ
m
(
g
)
|
+
∑
g
∈
S
ℓ
(
g
)
d
|
φ
m
(
g
)
−
φ
(
g
)
|
≤
‖
φ
n
−
φ
m
‖
d
+
∑
g
∈
S
ℓ
(
g
)
d
|
φ
m
(
g
)
−
φ
(
g
)
|
,
{\displaystyle {\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}
where the sum ranges over any finite subset
S
{\displaystyle S}
of
G
{\displaystyle G}
. Let
ϵ
>
0
{\displaystyle \epsilon >0}
, and let
K
ϵ
>
0
{\displaystyle K_{\epsilon }>0}
be such that
‖
φ
n
−
φ
m
‖
d
<
ϵ
{\displaystyle \|\varphi _{n}-\varphi _{m}\|_{d}<\epsilon }
for
m
,
n
≥
K
ϵ
{\displaystyle m,n\geq K_{\epsilon }
. By letting
m
{\displaystyle m}
run, we have
∑
g
∈
S
ℓ
(
g
)
d
|
φ
n
(
g
)
−
φ
(
g
)
|
<
ϵ
{\displaystyle \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon }
for
n
≥
K
ϵ
{\displaystyle n\geq K_{\epsilon }
. Summing over all of
G
{\displaystyle G}
, we therefore have
‖
φ
n
−
φ
‖
d
<
ϵ
{\displaystyle \left\|\varphi _{n}-\varphi \right\|_{d}<\epsilon }
for
n
≥
K
ϵ
{\displaystyle n\geq K_{\epsilon }
. By the estimate
∑
g
∈
S
ℓ
(
g
)
d
|
φ
(
g
)
|
≤
∑
g
∈
S
ℓ
(
g
)
d
|
φ
n
(
g
)
−
φ
(
g
)
|
+
∑
g
∈
S
ℓ
(
g
)
d
|
φ
n
(
g
)
|
≤
‖
φ
n
−
φ
‖
d
+
‖
φ
n
‖
d
,
{\displaystyle {\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}
we obtain
‖
φ
‖
d
<
∞
{\displaystyle \|\varphi \|_{d}<\infty }
. Since this holds for each
d
∈
N
{\displaystyle d\in \mathbb {N} }
, we have
φ
∈
A
{\displaystyle \varphi \in A}
and
φ
n
→
φ
{\displaystyle \varphi _{n}\to \varphi }
in the Fréchet topology, so
A
{\displaystyle A}
is complete.
^
‖
φ
∗
ψ
‖
d
≤
∑
g
∈
G
(
∑
h
∈
G
ℓ
(
g
)
d
|
φ
(
h
)
|
|
ψ
(
h
−
1
g
)
|
)
≤
∑
g
,
h
∈
G
(
ℓ
(
h
)
+
ℓ
(
h
−
1
g
)
)
d
|
φ
(
h
)
|
|
ψ
(
h
−
1
g
)
|
=
∑
i
=
0
d
(
d
i
)
(
∑
g
,
h
∈
G
|
ℓ
i
φ
(
h
)
|
|
ℓ
d
−
i
ψ
(
h
−
1
g
)
|
)
=
∑
i
=
0
d
(
d
i
)
(
∑
h
∈
G
|
ℓ
i
φ
(
h
)
|
)
(
∑
g
∈
G
|
ℓ
d
−
i
ψ
(
g
)
|
)
=
∑
i
=
0
d
(
d
i
)
‖
φ
‖
i
‖
ψ
‖
d
−
i
≤
2
d
‖
φ
‖
d
′
‖
ψ
‖
d
′
{\displaystyle {\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}
Citations
Sources
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Husain, Taqdir (1991). Orthogonal Schauder Bases . Pure and Applied Mathematics. Vol. 143. New York City: Marcel Dekker. ISBN 0-8247-8508-8 .
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Mitiagin, B.; Rolewicz, S.; Żelazko, W. (1962). "Entire functions in B 0 -algebras" . Studia Mathematica . 21 (3): 291–306. doi :10.4064/sm-21-3-291-306 . MR 0144222 .
Palmer, T.W. (1994). Banach Algebras and the General Theory of *-algebras, Volume I: Algebras and Banach Algebras . Encyclopedia of Mathematics and its Applications. Vol. 49. New York City: Cambridge University Press. ISBN 978-052136637-3 .
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Waelbroeck, Lucien (1971). Topological Vector Spaces and Algebras . Lecture Notes in Mathematics. Vol. 230. doi :10.1007/BFb0061234 . ISBN 978-354005650-8 . MR 0467234 .
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Żelazko, W. (1994). "Concerning entire functions in B 0 -algebras" . Studia Mathematica . 110 (3): 283–290. doi :10.4064/sm-110-3-283-290 . MR 1292849 .
Żelazko, W. (2001) [1994]. "Fréchet algebra". Encyclopedia of Mathematics . EMS Press.