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Non-Archimedean Riesz-Frechet Representation

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Abstract

We derive a Riesz-Frechet representation for bounded linear functionals defined on the p-adic Hilbert spaces introduced by Kalisch [Ann. of Math. (2), 1947]. We also notice the surprising difference between the Archimedean case and the non-Archimedean case (exact non-Archimedean version of Riesz-Frechet representation fails).

Introduction

The century-old Riesz-Frechet representation (theorem) states that every bounded linear functional on a real/complex Hilbert space is completely determined by a single element in that Hilbert space.

Theorem 1.1. [1, 2, 4, 8, 9, 10, 11, 12, 13, 14, 18, 19, 19, 20, 21, 23, 26, 28, 31, 32, 33, 34, 35, 36, 39, 43, 44, 46] (Riesz-Frechet Representation) Let $H$ be a complex Hilbert space. Let $f : H \to\mathbb{C}$ be a bounded linear functional. Then there is a unique element $\tau\in H$ satisfying the following conditions.

(i) $f(h) = \langle h, \tau\rangle$ for all $h \in H$ .

(ii) $\lVert f\rVert= \lVert\tau\rVert$ .

Over time, Riesz-Frechet representation has been extended to Lebesgue spaces [30, 37, 42] and spaces of continuous functionals on compact Hausdorff spaces (celebrated Riesz-Markov-Kakutani representation) [2, 3, 7, 8, 15, 24, 29, 35, 38, 45]. It is known that Riesz-Frechet representation also holds for quaternionic Hilbert spaces [16]. Riesz-Frechet representation has also been derived for certain classes of Banach spaces by Giles [17] and James [22]. The traditional proof of the Riesz-Frechet representation uses the orthogonal decomposition theorem, which rests on the parallelogram law [19, 32]. If the Hilbert space is separable, then one can also give a proof of the Riesz-Frechet representation for Hilbert spaces using orthonormal bases, which relies on the Gram-Schmidt orthonormalization process [20]. We are fundamentally motivated by the following question.

Question 1.2. What is the non-Archimedean Riesz-Frechet representation?

We begin by recalling the definition of non-Archimedean Banach and p-adic Hilbert spaces.

Definition 1.3. [40] Let $\mathbb{K}$ be a field. A map $\lvert\cdot\rvert: \mathbb{K} \to[0,\infty)$ is said to be a non-Archimedean valuation if the following conditions hold:

(i) If $\lambda\in\mathbb{K}$ satisfies $\lvert\lambda\rvert= 0$ , then $\lambda= 0$ .

(ii) $|\lambda\mu| = |\lambda||\mu|$ for all $\lambda, \mu\in\mathbb{K}$ .

(iii) (Ultratriangle inequality) $|\lambda+ \mu| \leq\max\{|\lambda|,|\mu|\}$ for all $\lambda, \mu\in\mathbb{K}$ .

We say $\mathbb{K}$ is a non-Archimedean complete valued field if it is complete w.r.t. the ultrametric

d(\lambda,\mu) := |\lambda-\mu|,\quad\forall\lambda,\mu\in\mathbb{K}.

Definition 1.4. [41] Let $\mathbb{K}$ be a non-Archimedean complete valued field. A vector space $\mathcal{X}$ over $\mathbb{K}$ is said to be non-Archimedean Banach space if there is a map (called as ultra-norm) $\|\cdot\|:\mathcal{X}\to[0,\infty)$ satisfying the following conditions:

(i) If $x\in\mathcal{X}$ is such that $\|x\|=0$ , then $x=0$ .

(ii) $\|\lambda x\|=|\lambda|\|x\|$ for all $\lambda\in\mathbb{K}$ , for all $x\in\mathcal{X}$ .

(iii) (Ultranorm inequality) $\|x+y\|\leq\max\{\|x\|,\|y\|\}$ for all $x,y\in\mathcal{X}$ .

(iv) $\mathcal{X}$ is complete w.r.t. the ultrametric

d(x,y):=\|x-y\|,\quad\forall x,y\in\mathcal{X}.

Definition 1.5. [25] Let $\mathbb{K}$ be a non-Archimedean valued field (with valuation $|\cdot|$ ) and $\mathcal{X}$ be a non-Archimedean Banach space (with norm $\|\cdot\|$ ) over $\mathbb{K}$ . We say that $\mathcal{X}$ is a $p$ -adic Hilbert space if there is a map (called as $p$ -adic inner product) $\langle\cdot,\cdot\rangle:\mathcal{X}\times\mathcal{X}\to\mathbb{K}$ satisfying the following conditions:

(i) If $x\in\mathcal{X}$ is such that $\langle x,y\rangle=0$ for all $y\in\mathcal{X}$ , then $x=0$ .

(ii) $\langle x,y\rangle=\langle y,x\rangle$ for all $x,y\in\mathcal{X}$ .

(iii) $\langle\alpha x,y\rangle=\alpha\langle x,y\rangle$ for all $\alpha\in\mathbb{K}$ , for all $x,y\in\mathcal{X}$ .

(iv) $\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$ for all $x,y,z\in\mathcal{X}$ .

(v) $|\langle x,y\rangle|\leq\|x\|\|y\|$ for all $x,y\in\mathcal{X}$ .

Definition 1.5 is more general than the definition of $p$ -adic Hilbert spaces used by Kalisch, which imposes further conditions [25]. We note that the Definition 1.5 is different than the $p$ -adic Hilbert spaces defined recently by Claussnitzer and Thom [6] where an exact Riesz-Frechet representation is known [5]. We derive two different versions of non-Archimedean Riesz-Frechet representation with different assumptions and different conclusions.

In the paper, we use the following notion of Schauder basis for non-Archimedean Banach spaces and orthonormal basis for $p$ -adic Hilbert spaces.

Definition 2.1. Let $\mathcal{X}$ be a non-Archimedean Banach space over $\mathbb{K}$ . A collection $\{\tau_n\}_n$ in $\mathcal{X}$ is said to be a Schauder basis for $\mathcal{X}$ if for every $x\in\mathcal{X}$ , there is a unique sequence $\{\alpha_n(x)\}_n$ in $\mathbb{K}$ such that

x=\sum_{n=1}^{\infty}\alpha_n(x)\tau_n,

series converges in the ultra-norm on $\mathcal{X}$ .

Definition 2.2. Let $\mathcal{X}$ be a $p$ -adic Hilbert space over $\mathbb{K}$ . A Schauder basis $\{\tau_n\}_n$ for $\mathcal{X}$ is said to be a $p$ -adic orthonormal basis for $\mathcal{X}$ if the following conditions hold:

(i) $\|\tau_n\|\leq1$ for all $n\in\mathbb{N}$ .

(ii) $\langle\tau_n,\tau_m\rangle=\delta_{n,m}$ for all $n,m\in\mathbb{N}$ .

Note that given a p-adic orthonormal basis $\{\tau_n\}_n$ for $\mathcal{X}$ , condition (ii) in Definition 2.2 says that

1 = \lvert\langle\tau_n,\tau_n\rangle\rvert\le\|\tau_n\|^2,\quad\forall n\in\mathbb{N}.

By using condition (i) in Definition 2.2 we then get $\|\tau_n\|=1$ for all $n\in\mathbb{N}$ . Thus all elements of p-adic orthonormal basis lie in the unit sphere. Following is the standard example of p-adic Hilbert space we keep in mind.

Example 2.3. Let $\mathbb{K}$ be a non-Archimedean valued field. Define

c_0(\mathbb{N},\mathbb{K}) := \left\{\{a_n\}_n : a_n\in\mathbb{K},\ \forall n\in\mathbb{N},\ \lim_{n\to\infty}a_n=0\right\}.

We define

\|\{a_n\}_n\| := \max_{n\in\mathbb{N}} |a_n|,\quad\forall\{a_n\}_n\in c_0(\mathbb{N},\mathbb{K})

and

\langle\{a_n\}_n,\{b_n\}_n\rangle:= \sum_{n=1}^{\infty}a_nb_n,\quad\forall\{a_n\}_n,\{b_n\}_n\in c_0(\mathbb{N},\mathbb{K}).

Then $c_0(\mathbb{N},\mathbb{K})$ is a p-adic Hilbert space. The canonical Schauder basis $\{e_n\}_n$ is a p-adic orthonormal basis for $c_0(\mathbb{N},\mathbb{K})$ .

Gram-Schmidt orthonormalization says that every separable Hilbert space has an orthonormal basis [27]. Eventhough we know from Example 2.3 that we have examples of p-adic orthonormal bases, we don’t know existence result of p-adic orthonormal basis in every p-adic Hilbert space. Following is our first version of Riesz-Frechet representation.

Theorem 2.4 (Non-Archimedean Riesz-Frechet Representation) Let $\mathcal{X}$ be a p-adic Hilbert space over a complete non-Archimedean valued field $\mathbb{K}$ . Let $f:\mathcal{X}\to\mathbb{K}$ be a bounded linear functional. Assume that there is a p-adic orthonormal basis $\{\omega_n\}_n$ for $\mathcal{X}$ such that

\lim_{n\to\infty}f(\omega_n)\omega_n=0.

Then there is a unique element $\tau\in\mathcal{X}$ satisfying the following conditions:

(i) $f(x)=\langle x,\tau\rangle$ for all $x\in\mathcal{X}$ .

(ii) $\|f\|=\|\tau\|$ .

Proof. Since $\{\omega_n\}_n$ is orthonormal,

\left\|\sum_{k=1}^{n}f(\omega_k)\omega_k\right\|=\max_{1\leq k\leq n}|f(\omega_k)|,\quad\forall n\in\mathbb{N}.

Since $\mathcal{X}$ is complete and $\lim_{n\to\infty}f(\omega_n)\omega_n=0$ , the series $\sum_{n=1}^{\infty}f(\omega_n)\omega_n$ converges in $\mathcal{X}$ . Now define

\tau:=\sum_{n=1}^{\infty}f(\omega_n)\omega_n.

We then have

\begin{aligned} f(x)&=f\left(\sum_{n=1}^{\infty}\langle x,\omega_n\rangle\omega_n\right)=\sum_{n=1}^{\infty}\langle x,\omega_n\rangle f(\omega_n)\\ &=\left\langle x,\sum_{n=1}^{\infty}f(\omega_n)\omega_n\right\rangle=\langle x,\tau\rangle,\quad\forall x\in\mathcal{X}. \end{aligned}

Now

\lVert f\rVert= \sup\left\{\frac{|f(x)|}{\lVert x\rVert}: x\in\mathcal{X}, x\ne0\right\}=\sup\left\{\frac{|\langle x,\tau\rangle|}{\lVert x\rVert}: x\in\mathcal{X}, x\ne0\right\}\leq\lVert\tau\rVert.

We now observe that

\lVert\tau\rVert=\left\lVert\sum_{n=1}^{\infty}f(\omega_n)\omega_n\right\rVert=\max_{n\in\mathbb{N}}|f(\omega_n)|\leq\lVert f\rVert\max_{n\in\mathbb{N}}\lVert\omega_n\rVert=\lVert f\rVert.

Uniqueness easily follows using the definition of p-adic inner product. □

Example 2.5. Let $p$ be a prime number. Define

f:c_0(\mathbb{N},\mathbb{Q}_p)\ni\{a_n\}_n\mapsto f(\{a_n\}_n):=\sum_{n=1}^{\infty}p^n a_n\in\mathbb{Q}_p.

Let $\{a_n\}_n\in c_0(\mathbb{N},\mathbb{Q}_p)$ . Then

|p^n a_n|_p=\frac{1}{p^n}|a_n|_p\to0\quad\text{as }n\to\infty.

Hence

\lim_{n\to\infty}p^n a_n=0,\quad\forall\{a_n\}_n\in c_0(\mathbb{N},\mathbb{Q}_p).

So the map $f$ is well-defined. Define

\tau:=\{p^n\}_n\in c_0(\mathbb{N},\mathbb{Q}_p).

Then

f(x)=\langle x,\tau\rangle,\ \forall x\in c_0(\mathbb{N},\mathbb{Q}_p)\quad\text{and}\quad\lVert f\rVert=\lVert\tau\rVert=\max_{n\in\mathbb{N}}|p^n|_p=\max_{n\in\mathbb{N}}\left\{\frac{1}{p^n}\right\}=\frac{1}{p}.

In the next example we show that every bounded linear functional on a p-adic Hilbert space cannot be represented by an element in the space.

Example 2.6. Let $p$ be a prime number. Define

f:c_0(\mathbb{N},\mathbb{Q}_p)\ni\{a_n\}_n\mapsto f(\{a_n\}_n):=\sum_{n=1}^{\infty}na_n\in\mathbb{Q}_p.

Let $\{a_n\}_n\in c_0(\mathbb{N},\mathbb{Q}_p)$ . Then

|na_n|_p=|n||a_n|_p\leq|a_n|_p\to0\quad\text{as }n\to\infty.

Hence

\lim_{n\to\infty}na_n=0,\quad\forall\{a_n\}_n\in c_0(\mathbb{N},\mathbb{Q}_p).

So the map $f$ is well-defined. We claim that there is no $\tau\in c_0(\mathbb{N},\mathbb{Q}_p)$ such that $f(x)=\langle x,\tau\rangle$ for all $x\in c_0(\mathbb{N},\mathbb{Q}_p)$ . Let us suppose that this claim fails. Then thee exists a $\tau\in c_0(\mathbb{N},\mathbb{Q}_p)$ such that $f(x)=\langle x,\tau\rangle$ for all $x\in c_0(\mathbb{N},\mathbb{Q}_p)$ . Set $\tau:=\{a_n\}_n\in c_0(\mathbb{N},\mathbb{Q}_p)$ . Let $\{e_n\}$ be the standard p-adic orthonormal basis for $c_0(\mathbb{N},\mathbb{Q}_p)$ . Now we see that

n=f(e_n)=\langle e_n,\tau\rangle=a_n,\quad\forall n\in\mathbb{N}.

But then $\tau=\{n\}_n\notin c_0(\mathbb{N},\mathbb{Q}_p)$ because $|n|_p\nrightarrow0$ as $n\to\infty$ . Hence Riesz-Frechet representation fails.

Let $\mathcal{X}$ be a p-adic Hilbert space. A closed subspace $\mathcal{Y}$ of $\mathcal{X}$ is said to be orthogonally complementable if there exists a unique closed subspace $\mathcal{Z}$ of $\mathcal{X}$ satisfying the following: for every $x \in\mathcal{X}$ , there exist unique $y \in\mathcal{Y}$ and $z \in\mathcal{Z}$ such that

(i) $x = y + z$ .

(ii) $\langle y,z\rangle= 0$ .

(iii) $\lVert x\rVert= \max\{\lVert y\rVert,\lVert z\rVert\}$ .

In this case, we write $x = y \oplus z$ , $\mathcal{Z} = \mathcal{Y}^{\perp}$ and $\mathcal{X} = \mathcal{Y} \oplus\mathcal{Y}^{\perp}$ . Following is our second version of Riesz-Frechet representation.

Theorem 2.7. (Non-Archimedean Riesz-Frechet Representation) Let $\mathcal{X}$ be a p-adic Hilbert space over a complete non-Archimedean valued field $\mathbb{K}$ . Let $f:\mathcal{X} \to\mathbb{K}$ be a non-zero bounded linear functional. Assume that $\operatorname{Ker}(f)$ is orthogonally complementable in $\mathcal{X}$ and there exists a (nonzero) vector $\omega\in\operatorname{Ker}(f)^{\perp}$ such that $\langle\omega,\omega\rangle\ne0$ . Then there is a unique element $\tau\in\mathcal{X}$ satisfying the following conditions:

(i) $f(x) = \langle x,\tau\rangle$ for all $x \in\mathcal{X}$ .

(ii)

\frac{\lvert\langle\tau,\tau\rangle\rvert}{\lVert\tau\rVert} \le\lVert f\rVert\le\lVert\tau\rVert.

Moreover,

\lVert\tau\rVert= \frac{\lvert f(\omega)\rvert}{\lvert\langle\omega,\omega\rangle\rvert}\lVert\omega\rVert\le\frac{\lVert f\rVert\lVert\omega\rVert^2}{\lvert\langle\omega,\omega\rangle\rvert}.

Proof. Define

\tau:= \frac{f(\omega)}{\langle\omega,\omega\rangle}\omega.

Let $x \in\mathcal{X}$ . Then

f\left(x-\frac{f(x)}{f(\omega)}\omega\right)=f(x)-f(x)=0.

Hence

x-\frac{f(x)}{f(\omega)}\omega\in\operatorname{Ker}(f).

We then have

\begin{aligned} 0=\left\langle x-\frac{f(x)}{f(\omega)}\omega,\tau\right\rangle =\langle x,\tau\rangle-\left\langle\frac{f(x)}{f(\omega)}\omega,\tau\right\rangle\\ &=\langle x,\tau\rangle-\frac{f(x)}{f(\omega)}\langle\omega,\tau\rangle =\langle x,\tau\rangle-\frac{f(x)}{f(\omega)} \left\langle\omega,\frac{f(\omega)}{\langle\omega,\omega\rangle}\omega\right\rangle\\ &=\langle x,\tau\rangle-f(x). \end{aligned}

Note that

\lvert\langle\tau,\tau\rangle\rvert=\lvert f(\tau)\rvert\le\lVert f\rVert\lVert\tau\rVert.

□

We next extend the non-Archimedean Riesz-Frechet representation to non-Archimedean fields admitting conjugation (involution). We use the following definition of conjugation.

Definition 2.8. Let $\mathbb{K}$ be a non-Archimedean complete valued field (with valuation $|\cdot|$ ). A map $\sigma:\mathbb{K}\to\mathbb{K}$ is said to be a conjugation or involution if the following conditions hold:

(i) $\sigma(\alpha+\beta)=\sigma(\alpha)+\sigma(\beta)$ for all $\alpha,\beta\in\mathbb{K}$ .

(ii) $\sigma(\alpha\beta)=\sigma(\alpha)\sigma(\beta)$ for all $\alpha,\beta\in\mathbb{K}$ .

(iii) $\sigma(\sigma(\alpha))=\alpha$ for all $\alpha\in\mathbb{K}$ .

(iv) $|\sigma(\alpha)|=|\alpha|$ for all $\alpha\in\mathbb{K}$ .

In this case, we say that $\mathbb{K}$ is $\sigma$ -conjugated.

Definition 2.9. Let $\mathbb{K}$ be a $\sigma$ -conjugated non-Archimedean complete valued field (with valuation $|\cdot|$ ) and $\mathcal{X}$ be a non-Archimedean Banach space (with norm $\|\cdot\|$ ) over $\mathbb{K}$ . We say that $\mathcal{X}$ is a p-adic $\sigma$ -Hilbert space if there is a map (called as p-adic $\sigma$ -inner product) $\langle\cdot,\cdot\rangle:\mathcal{X}\times\mathcal{X}\to\mathbb{K}$ satisfying the following:

(i) If $x\in\mathcal{X}$ is such that $\langle x,y\rangle=0$ for all $y\in\mathcal{X}$ , then $x=0$ .

(ii) $\langle x,y\rangle=\sigma(\langle y,x\rangle)$ for all $x,y\in\mathcal{X}$ .

(iii) $\langle\alpha x,y\rangle=\alpha\langle x,y\rangle$ for all $\alpha\in\mathbb{K}$ , for all $x,y\in\mathcal{X}$ .

(iv) $\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$ for all $x,y,z\in\mathcal{X}$ .

(v) $|\langle x,y\rangle|\leq\|x\|\|y\|$ for all $x,y\in\mathcal{X}$ .

We now have p-adic versions of Theorem 2.4 and Theorem 2.7.

Theorem 2.10. (Non-Archimedean Riesz-Frechet Representation) Let $\mathcal{X}$ be a p-adic $\sigma$ -Hilbert space over a $\sigma$ -conjugated complete non-Archimedean valued field $\mathbb{K}$ . Let $f:\mathcal{X}\to\mathbb{K}$ be a bounded linear functional. Assume that there is a p-adic orthonormal basis $\{\omega_n\}_n$ for $\mathcal{X}$ such that

\lim_{n\to\infty} f(\omega_n)\omega_n=0.

Then there is a unique element $\tau\in\mathcal{X}$ satisfying the following conditions:

(i) $f(x)=\langle x,\tau\rangle$ for all $x\in\mathcal{X}$ .

(ii) $\|f\|=\|\tau\|$ .

Proof. Define

\tau:=\sum_{n=1}^{\infty}\sigma(f(\omega_n))\omega_n.

The other arguments are similar to those used in the proof of Theorem 2.4. □

Theorem 2.11. (Non-Archimedean Riesz-Frechet Representation) Let $\mathcal{X}$ be a p-adic $\sigma$ -Hilbert space over a $\sigma$ -conjugated complete non-Archimedean valued field $\mathbb{K}$ . Let $f:\mathcal{X}\to\mathbb{K}$ be a non-zero bounded linear functional. Assume that $\operatorname{Ker}(f)$ is orthogonally complementable in $\mathcal{X}$ and there exists a (nonzero) vector $\omega\in\operatorname{Ker}(f)^\perp$ such that $\langle\omega,\omega\rangle\ne0$ . Then there is a unique element $\tau\in\mathcal{X}$ satisfying the following conditions:

(i) $f(x)=\langle x,\tau\rangle$ for all $x\in\mathcal{X}$ .

(ii)

\frac{|\langle\tau,\tau\rangle|}{\|\tau\|}\leq\|f\|\leq\|\tau\|.

Moreover,

\|\tau\|=\frac{|f(\omega)|}{|\langle\omega,\omega\rangle|}\|\omega\|\leq\frac{\|f\|\|\omega\|^2}{|\langle\omega,\omega\rangle|}.

Proof. Define

\tau:= \frac{\sigma(f(\omega))}{\langle\omega,\omega\rangle}\omega.

The other arguments are similar to those used in the proof of Theorem 2.7.

Conclusions

In 1907, Riesz [34] and Frechet [13] independently showed that every continuous linear functional on a Hilbert space is completely determined by a single element in that Hilbert space [4].

In 1938, Markov [29] and in 1941, Kakutani [24] extended the Riesz-Frechet representation to function spaces.

In 1947, James [22] and 1967, Giles [17] extended the Riesz-Frechet representation to certain classes of Banach spaces.

In this article, we derived two non-Archimedean Riesz representation theorems under certain conditions for $p$ -adic Hilbert spaces.

References

[1]N. I. Akhiezer and I. M. Glazman. Theory of linear operators in Hilbert space. New York, NY: Dover Publications, 1993.
[2]Charalambos D. Aliprantis and Kim C. Border. Infinite dimensional analysis: A hitchhiker’s guide. Springer, 2006.
[3]George Bachman and Lawrence Narici. Functional analysis. Dover Publications, New York, 2000.
[4]Garrett Birkhoff and Erwin Kreyszig. The establishment of functional analysis. Hist. Math., 11:258–321, 1984.DOI
[5]Alcides Buss, Luiz Felipe Garcia, and Devarshi Mukherjee. Operator algebras over the p-adic integers. Trans. Am. Math. Soc., 378(10):7371–7427, 2025.arxiv.org/abs/2403.04046
[6]Anton Claussnitzer and Andreas Thom. Aspects of p-adic operator algebras. Münster J. Math., 13(2):425–444, 2020.DOI
[7]Donald L. Cohn. Measure theory. Birkhäuser Adv. Texts, Basler Lehrbüch. New York, NY: Birkhäuser/Springer, 2013.
[8]John B. Conway. A course in functional analysis. Springer-Verlag, 1990.
[9]Lokenath Debnath and Piotr Mikusinski. Introduction to Hilbert spaces with applications. Academic Press, 1999.
[10]Jean Dieudonne. History of functional analysis, volume 49 of North-Holland Math. Stud. Elsevier, Amsterdam, 1981.
[11]Ronald G. Douglas. Banach algebra techniques in operator theory. Springer, 1998.DOI
[12]Nelson Dunford and Jacob T. Schwartz. Linear operators, I, General theory. Interscience Publishers, 1958.
[13]M. Frechet. Sur les operations lineaires. Trans. Am. Math. Soc., 8:433–446, 1907.
[14]Paul A. Fuhrmann. Linear systems and operators in Hilbert space. McGraw-Hill International Book Company, 1981.
[15]D. J. H. Garling. A ’short’ proof of the Riesz representation theorem. Proc. Camb. Philos. Soc., 73:459–460, 1973.DOI
[16]Riccardo Ghiloni, Valter Moretti, and Alessandro Perotti. Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys., 25(4):83, 2013.DOI
[17]J. R. Giles. Classes of semi-inner-product spaces. Trans. Am. Math. Soc., 129:436–446, 1967.DOI
[18]J. D. Gray. The shaping of the Riesz representation theorem: A chapter in the history of analysis. Arch. Hist. Exact Sci., 31:127–187, 1984.DOI
[19]P. R. Halmos. Introduction to Hilbert space and the theory of spectral multiplicity. Chelsea Publishing Company, New York, 1957.
[20]Paul R. Halmos. A Hilbert space problem book. Springer, 1982.
[21]Vagn Lundsgaard Hansen. Functional analysis. Entering Hilbert space. Hackensack, NJ: World Scientific, 2016.
[22]Robert C. James. Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc., 61:285–292, 1947.DOI
[23]Richard V. Kadison and John R. Ringrose. Fundamentals of the theory of operator algebras, Vol. 1. Academic Press, San Diego, 1994.
[24]Shizuo Kakutani. Concrete representation of abstract (M)-spaces (A characterization of the space of continuous functions). Ann. Math. (2), 42:994–1024, 1941.DOI
[25]G. K. Kalisch. On p-adic Hilbert spaces. Ann. of Math. (2), 48:180–192, 1947.DOI
[26]Erwin Kreyszig. Introductory functional analysis with applications. John Wiley & Sons, New York, 1978.
[27]Steven J. Leon, Ake Bjorck, and Walter Gander. Gram-Schmidt orthogonalization: 100 years and more. Numer. Linear Algebra Appl., 20(3):492–532, 2013.DOI
[28]Balmohan V. Limaye. Functional analysis. New Delhi: New Age International Limited, 1996.
[29]A. A. Markov. On mean values and exterior densities. Rec. Math. Moscou, n. Ser., 4:165–191, 1938.
[30]E. J. McShane. Linear functionals on certain Banach spaces. Proc. Am. Math. Math. Soc., 1:402–408, 1950.
[31]Joseph Muscat. Functional analysis. An introduction to metric spaces, Hilbert spaces, and Banach algebras. Springer, 2024.
[32]Marco Pavone. On the Riesz representation theorem for bounded linear functionals. Proc. R. Ir. Acad., Sect. A, 94(1):133–135, 1994.
[33]Albrecht Pietsch. History of Banach spaces and linear operators. Basel: Birkhäuser, 2007.
[34]F. Riesz. Sur une espece de geometrie analytique des systemes de fonctions sommables. C. R. Acad. Sci., Paris, 144:1409–1411, 1907.
[35]Friedrich Riesz. Zur Theorie des Hilbertschen Raumes. Acta Litt. Sci. Szeged, 7:34–38, 1934.
[36]Frigyes Riesz and Bela Sz.-Nagy. Functional analysis. London-Glasgow: Blackie & Son, Ltd, 1956.
[37]Halsey Royden and Patrick M. Fitzpatrick. Real analysis. Prentice Hall, New York, 2010.
[38]Walter Rudin. Real and complex analysis. McGraw-Hill, 1987.
[39]Walter Rudin. Functional analysis. McGraw-Hill, 1991.
[40]W. H. Schikhof. Ultrametric calculus. An introduction to p-adic analysis. Cambridge University Press, Cambridge, 1984.
[41]Peter Schneider. Nonarchimedean functional analysis. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002.DOI
[42]Naoki Shioji. Simple proofs of the uniform convexity of Lp and the Riesz representation theorem for Lp. Am. Math. Mon., 125(8):733–738, 2018.
[43]Marshall Harvey Stone. Linear transformations in Hilbert space and their applications to analysis, volume 15 of Colloq. Publ., Am. Math. Soc. American Mathematical Society (AMS), Providence, RI, 1932.
[44]Joachim Weidmann. Linear operators in Hilbert spaces. Springer, 1980.DOI
[45]J. Yeh. Real analysis. Theory of measure and integration. World Scientific, 2014.
[46]Nicholas Young. An introduction to Hilbert space. Cambridge University Press, 1988.

Non-Archimedean Riesz-Frechet Representation

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