Fréchet algebra

In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra $A$ over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation $(a,b)\mapsto a*b$ for $a,b\in A$ is required to be jointly continuous. If $\{\|\cdot \|_{n}\}_{n=0}^{\infty }$ is an increasing family^[a] of seminorms for the topology of $A$ , the joint continuity of multiplication is equivalent to there being a constant $C_{n}>0$ and integer $m\geq n$ for each $n$ such that $\left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}$ for all $a,b\in A$ .^[b] Fréchet algebras are also called B₀-algebras.^[1]

A Fréchet algebra is $m$ -convex if there exists such a family of semi-norms for which $m=n$ . In that case, by rescaling the seminorms, we may also take $C_{n}=1$ for each $n$ and the seminorms are said to be submultiplicative: $\|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}$ for all $a,b\in A.$ ^[c] $m$ -convex Fréchet algebras may also be called Fréchet algebras.^[2]

A Fréchet algebra may or may not have an identity element $1_{A}$ . If $A$ is unital, we do not require that $\|1_{A}\|_{n}=1,$ as is often done for Banach algebras.

Properties

Continuity of multiplication. Multiplication is separately continuous if $a_{k}b\to ab$ and $ba_{k}\to ba$ for every $a,b\in A$ and sequence $a_{k}\to a$ converging in the Fréchet topology of $A$ . Multiplication is jointly continuous if $a_{k}\to a$ and $b_{k}\to b$ imply $a_{k}b_{k}\to ab$ . 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 $invA$ is the set of invertible elements of $A$ , then the inverse map ${\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}}$ is continuous if and only if $invA$ is a $G_{\delta }$ set.^[4] Unlike for Banach algebras, $invA$ may not be an open set. If $invA$ is open, then $A$ is called a $Q$ -algebra. (If $A$ happens to be non-unital, then we may adjoin a unit to $A$ ^[d] and work with $invA^{+}$ , or the set of quasi invertibles^[e] may take the place of $invA$ .)
Conditions for $m$ -convexity. A Fréchet algebra is $m$ -convex if and only if for every, if and only if for one, increasing family $\{\|\cdot \|_{n}\}_{n=0}^{\infty }$ of seminorms which topologize $A$ , for each $m\in \mathbb {N}$ there exists $p\geq m$ and $C_{m}>0$ such that $\|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},$ for all $a_{1},a_{2},\dots ,a_{n}\in A$ and $n\in \mathbb {N}$ .^[5] A commutative Fréchet $Q$ -algebra is $m$ -convex,^[6] but there exist examples of non-commutative Fréchet $Q$ -algebras which are not $m$ -convex.^[7]
Properties of $m$ -convex Fréchet algebras. A Fréchet algebra is $m$ -convex if and only if it is a countable projective limit of Banach algebras.^[8] An element of $A$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible.^[f]^[9]^[10]

Examples

Zero multiplication. If $E$ is any Fréchet space, we can make a Fréchet algebra structure by setting $e*f=0$ for all $e,f\in E$ .
Smooth functions on the circle. Let $S^{1}$ be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let $A=C^{\infty }(S^{1})$ be the set of infinitely differentiable complex-valued functions on $S^{1}$ . 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 $1$ acts as an identity. Define a countable set of seminorms on $A$ by $\left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,$ where $\left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}}}\left|\varphi ^{(n)}(x)\right|$ denotes the supremum of the absolute value of the $n$ th derivative $\varphi ^{(n)}$ .^[g] Then, by the product rule for differentiation, we have ${\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}}$ where ${n \choose i}={\frac {n!}{i!(n-i)!}},$ denotes the binomial coefficient and $\|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.$ The primed seminorms are submultiplicative after re-scaling by $C_{n}=2^{n}$ .
Sequences on $\mathbb {N}$ . Let $\mathbb {C} ^{\mathbb {N} }$ be the space of complex-valued sequences on the natural numbers $\mathbb {N}$ . Define an increasing family of seminorms on $\mathbb {C} ^{\mathbb {N} }$ by $\|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.$ With pointwise multiplication, $\mathbb {C} ^{\mathbb {N} }$ is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative $\|\varphi \psi \|_{n}\leq \|\varphi \|_{n}\|\psi \|_{n}$ for $\varphi ,\psi \in A$ . This $m$ -convex Fréchet algebra is unital, since the constant sequence $1(k)=1,k\in \mathbb {N}$ is in $A$ .
Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, $C(\mathbb {C} )$ , the algebra of all continuous functions on the complex plane $\mathbb {C}$ , or to the algebra $\mathrm {Hol} (\mathbb {C} )$ of holomorphic functions on $\mathbb {C}$ .
Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let $G$ be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements $U=\{g_{1},\dots ,g_{n}\}\subseteq G$ such that: $\bigcup _{n=0}^{\infty }U^{n}=G.$ Without loss of generality, we may also assume that the identity element $e$ of $G$ is contained in $U$ . Define a function $\ell :G\to [0,\infty )$ by $\ell (g)=\min\{n\mid g\in U^{n}\}.$ Then $\ell (gh)\leq \ell (g)+\ell (h)$ , and $\ell (e)=0$ , since we define $U^{0}=\{e\}$ .^[h] Let $A$ be the $\mathbb {C}$ -vector space $S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \}},$ where the seminorms $\|\cdot \|_{d}$ are defined by $\|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.$ ^[i] $A$ is an $m$ -convex Fréchet algebra for the convolution multiplication $\varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),$ ^[j] $A$ is unital because $G$ is discrete, and $A$ is commutative if and only if $G$ is Abelian.
Non $m$ -convex Fréchet algebras. The Aren's algebra $A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]$ is an example of a commutative non- $m$ -convex Fréchet algebra with discontinuous inversion. The topology is given by $L^{p}$ norms $\|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,$ and multiplication is given by convolution of functions with respect to Lebesgue measure on $[0,1]$ .^[11]

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^[12] or an F-space.^[13]

If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC).^[14] A complete LMC algebra is called an Arens-Michael algebra.^[15]

Michael's Conjecture

The question of whether all linear multiplicative functionals on an $m$ -convex Frechet algebra are continuous is known as Michael's Conjecture.^[16]. For a long time, this conjecture was perhaps the most famous open problem in the theory of topological algebras. Michael's Conjecture was solved completely and affirmatively in 2022.^[17]

Notes

^ An increasing family means that for each $a\in A,$
$\|a\|_{0}\leq \|a\|_{1}\leq \cdots \leq \|a\|_{n}\leq \cdots$ .
^ Joint continuity of multiplication means that for every absolutely convex neighborhood $V$ of zero, there is an absolutely convex neighborhood $U$ of zero for which $U^{2}\subseteq V,$ from which the seminorm inequality follows. Conversely,
${\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}}$
^ In other words, an $m$ -convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: $p(fg)\leq p(f)p(g),$ and the algebra is complete.
^ If $A$ is an algebra over a field $k$ , the unitization $A^{+}$ of $A$ is the direct sum $A\oplus k1$ , with multiplication defined as $(a+\mu 1)(b+\lambda 1)=ab+\mu b+\lambda a+\mu \lambda 1.$
^ If $a\in A$ , then $b\in A$ is a quasi-inverse for $a$ if $a+b-ab=0$ .
^ If $A$ is non-unital, replace invertible with quasi-invertible.
^ To see the completeness, let $\varphi _{k}$ be a Cauchy sequence. Then each derivative $\varphi _{k}^{(l)}$ is a Cauchy sequence in the sup norm on $S^{1}$ , and hence converges uniformly to a continuous function $\psi _{l}$ on $S^{1}$ . It suffices to check that $\psi _{l}$ is the $l$ th derivative of $\psi _{0}$ . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have
${\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}}^{x}\psi _{l+1}(t)dt.\end{aligned}}$
^ We can replace the generating set $U$ with $U\cup U^{-1}$ , so that $U=U^{-1}$ . Then $\ell$ satisfies the additional property $\ell (g^{-1})=\ell (g)$ , and is a length function on $G$ .
^ To see that $A$ is Fréchet space, let $\varphi _{n}$ be a Cauchy sequence. Then for each $g\in G$ , $\varphi _{n}(g)$ is a Cauchy sequence in $\mathbb {C}$ . Define $\varphi (g)$ to be the limit. Then
${\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}}$
where the sum ranges over any finite subset $S$ of $G$ . Let $\epsilon >0$ , and let $K_{\epsilon }>0$ be such that $\|\varphi _{n}-\varphi _{m}\|_{d}<\epsilon$ for $m,n\geq K_{\epsilon }$ . By letting $m$ run, we have
$\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon$
for $n\geq K_{\epsilon }$ . Summing over all of $G$ , we therefore have $\left\|\varphi _{n}-\varphi \right\|_{d}<\epsilon$ for $n\geq K_{\epsilon }$ . By the estimate
${\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}}$
we obtain $\|\varphi \|_{d}<\infty$ . Since this holds for each $d\in \mathbb {N}$ , we have $\varphi \in A$ and $\varphi _{n}\to \varphi$ in the Fréchet topology, so $A$ is complete.
^
${\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}}$

Citations

^ Mitiagin, Rolewicz & Żelazko 1962; Żelazko 2001.
^ Husain 1991; Żelazko 2001.
^ Waelbroeck 1971, Chapter VII, Proposition 1; Palmer 1994, $\S$ 2.9.
^ Waelbroeck 1971, Chapter VII, Proposition 2.
^ Mitiagin, Rolewicz & Żelazko 1962, Lemma 1.2.
^ Żelazko 1965, Theorem 13.17.
^ Żelazko 1994, pp. 283–290.
^ Michael 1952, Theorem 5.1.
^ Michael 1952, Theorem 5.2.
^ See also Palmer 1994, Theorem 2.9.6.
^ Fragoulopoulou 2005, Example 6.13 (2).
^ Waelbroeck 1971.
^ Rudin 1973, 1.8(e).
^ Michael 1952; Husain 1991.
^ Fragoulopoulou 2005, Chapter 1.
^ Michael 1952, $\S$ 12, Question 1; Palmer 1994, $\S$ 3.1.
^ Patel, S. R. (2022-06-28), On affirmative solution to Michael's acclaimed problem in the theory of Fr\'{e}chet algebras, with applications to automatic continuity theory, doi:10.48550/arXiv.2006.11134, retrieved 2024-07-11

Sources

Fragoulopoulou, Maria (2005). Topological Algebras with Involution. North-Holland Mathematics Studies. Vol. 200. Amsterdam: Elsevier B.V. doi:10.1016/S0304-0208(05)80031-3. ISBN 978-044452025-8.
Husain, Taqdir (1991). Orthogonal Schauder Bases. Pure and Applied Mathematics. Vol. 143. New York City: Marcel Dekker. ISBN 0-8247-8508-8.
Michael, Ernest A. (1952). Locally Multiplicatively-Convex Topological Algebras. Memoirs of the American Mathematical Society. Vol. 11. MR 0051444.
Mitiagin, B.; Rolewicz, S.; Żelazko, W. (1962). "Entire functions in B₀-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.
Rudin, Walter (1973). Functional Analysis. Series in Higher Mathematics. New York City: McGraw-Hill Book. 1.8(e). ISBN 978-007054236-5 – via Internet Archive.
Waelbroeck, Lucien (1971). Topological Vector Spaces and Algebras. Lecture Notes in Mathematics. Vol. 230. doi:10.1007/BFb0061234. ISBN 978-354005650-8. MR 0467234.
Żelazko, W. (1965). "Metric generalizations of Banach algebras". Rozprawy Mat. (Dissertationes Math.). 47. Theorem 13.17. MR 0193532.
Żelazko, W. (1994). "Concerning entire functions in B₀-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.

[1] An increasing family means that for each $a\in A,$
$\|a\|_{0}\leq \|a\|_{1}\leq \cdots \leq \|a\|_{n}\leq \cdots$ .

[2] Joint continuity of multiplication means that for every absolutely convex neighborhood $V$ of zero, there is an absolutely convex neighborhood $U$ of zero for which $U^{2}\subseteq V,$ from which the seminorm inequality follows. Conversely,
${\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}}$

[4] In other words, an $m$ -convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: $p(fg)\leq p(f)p(g),$ and the algebra is complete.

[8] If $A$ is an algebra over a field $k$ , the unitization $A^{+}$ of $A$ is the direct sum $A\oplus k1$ , with multiplication defined as $(a+\mu 1)(b+\lambda 1)=ab+\mu b+\lambda a+\mu \lambda 1.$

[9] If $a\in A$ , then $b\in A$ is a quasi-inverse for $a$ if $a+b-ab=0$ .

[14] If $A$ is non-unital, replace invertible with quasi-invertible.

[17] To see the completeness, let $\varphi _{k}$ be a Cauchy sequence. Then each derivative $\varphi _{k}^{(l)}$ is a Cauchy sequence in the sup norm on $S^{1}$ , and hence converges uniformly to a continuous function $\psi _{l}$ on $S^{1}$ . It suffices to check that $\psi _{l}$ is the $l$ th derivative of $\psi _{0}$ . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have
${\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}}^{x}\psi _{l+1}(t)dt.\end{aligned}}$

[18] We can replace the generating set $U$ with $U\cup U^{-1}$ , so that $U=U^{-1}$ . Then $\ell$ satisfies the additional property $\ell (g^{-1})=\ell (g)$ , and is a length function on $G$ .

[19] To see that $A$ is Fréchet space, let $\varphi _{n}$ be a Cauchy sequence. Then for each $g\in G$ , $\varphi _{n}(g)$ is a Cauchy sequence in $\mathbb {C}$ . Define $\varphi (g)$ to be the limit. Then
${\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}}$
where the sum ranges over any finite subset $S$ of $G$ . Let $\epsilon >0$ , and let $K_{\epsilon }>0$ be such that $\|\varphi _{n}-\varphi _{m}\|_{d}<\epsilon$ for $m,n\geq K_{\epsilon }$ . By letting $m$ run, we have
$\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon$
for $n\geq K_{\epsilon }$ . Summing over all of $G$ , we therefore have $\left\|\varphi _{n}-\varphi \right\|_{d}<\epsilon$ for $n\geq K_{\epsilon }$ . By the estimate
${\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}}$
we obtain $\|\varphi \|_{d}<\infty$ . Since this holds for each $d\in \mathbb {N}$ , we have $\varphi \in A$ and $\varphi _{n}\to \varphi$ in the Fréchet topology, so $A$ is complete.

[20] 
${\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}}$

[FOOTNOTEMitiaginRolewiczŻelazko1962Żelazko2001-3] Mitiagin, Rolewicz & Żelazko 1962; Żelazko 2001.

[5] Husain 1991; Żelazko 2001.

[6] Waelbroeck 1971, Chapter VII, Proposition 1; Palmer 1994, $\S$ 2.9.

[FOOTNOTEWaelbroeck1971Chapter_VII,_Proposition_2-7] Waelbroeck 1971, Chapter VII, Proposition 2.

[FOOTNOTEMitiaginRolewiczŻelazko1962Lemma_1.2-10] Mitiagin, Rolewicz & Żelazko 1962, Lemma 1.2.

[FOOTNOTEŻelazko1965Theorem_13.17-11] Żelazko 1965, Theorem 13.17.

[FOOTNOTEŻelazko1994283–290-12] Żelazko 1994, pp. 283–290.

[FOOTNOTEMichael1952Theorem_5.1-13] Michael 1952, Theorem 5.1.

[FOOTNOTEMichael1952Theorem_5.2-15] Michael 1952, Theorem 5.2.

[16] See also Palmer 1994, Theorem 2.9.6.

[FOOTNOTEFragoulopoulou2005Example_6.13_(2)-21] Fragoulopoulou 2005, Example 6.13 (2).

[FOOTNOTEWaelbroeck1971-22] Waelbroeck 1971.

[FOOTNOTERudin19731.8(e)-23] Rudin 1973, 1.8(e).

[FOOTNOTEMichael1952Husain1991-24] Michael 1952; Husain 1991.

[FOOTNOTEFragoulopoulou2005Chapter_1-25] Fragoulopoulou 2005, Chapter 1.

[26] Michael 1952, $\S$ 12, Question 1; Palmer 1994, $\S$ 3.1.

[27] Patel, S. R. (2022-06-28), On affirmative solution to Michael's acclaimed problem in the theory of Fr\'{e}chet algebras, with applications to automatic continuity theory, doi:10.48550/arXiv.2006.11134, retrieved 2024-07-11

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