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\fancyhead[LO]{\textbf{Non-Archimedean Riesz-Frechet Representation}}
\fancyhead[RE]{\textbf{K. MAHESH KRISHNA}}


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{\bf{Non-Archimedean Riesz-Frechet Representation}}\\
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\textbf{K. MAHESH KRISHNA}\\
School of Mathematics and Natural Sciences\\
Chanakya University Global Campus\\
NH-648, Haraluru Village\\
Devanahalli Taluk, 	Bengaluru  North District\\
Karnataka State 562 110 India\\
Email: kmaheshak@gmail.com\\

Date: \today
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\textbf{Abstract}: We derive a Riesz-Frechet representation for bounded linear functionals defined on the p-adic Hilbert spaces introduced by Kalisch  [\textit{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).


\textbf{Keywords}: Hilbert Space, Riesz-Frechet representation, p-adic Hilbert space.

\textbf{Mathematics Subject Classification (2020)}: 46S10, 46C05. \\

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\section{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.
\begin{theorem}\cite{KREYSZIG, DEBNATHMIKUSINSKI, DIEUDONNE, BIRKHOFFKREYSZIG, GRAY, FRECHET, RIESZ, RIESZ2, HALMOS, HALMOS2,  PAVONE, HALMOS2, DOUGLAS, RUDIN, KADISONRINGROSE, YOUNG, DUNFORDSCHWARTZ, RIESZNAGY, LIMAYE, PIETSCH, STONE, HANSEN, MUSCAT, AKHIEZERGLAZMAN, WEIDMANN, FUHRMANN, CONWAY, ALIPRANTISBORDER}
	(\textbf{Riesz-Frechet Representation}) Let $\mathcal{H}$ be a complex Hilbert space. Let $f:\mathcal{H} \to \mathbb{C}$ be a bounded linear functional. Then there is a unique element $\tau \in \mathcal{H}$ satisfying the following conditions. 
\begin{enumerate}[\upshape(i)]
	\item $f(h)=\langle h, \tau \rangle$ for all $h \in \mathcal{H}$.
	\item $\|f\|=\|\tau\|$.
\end{enumerate}
\end{theorem}
Over time, Riesz-Frechet representation has been extended to Lebesgue spaces \cite{MCSHANE, SHIOJI, ROYDEN} and spaces of continuous functionals on compact Hausdorff spaces (celebrated Riesz-Markov-Kakutani representation) \cite{BACHMANNARICI, RIESZ2, KAKUTANI, MARKOV, GARLING, RUDIN2, YEH, COHN, CONWAY, ALIPRANTISBORDER}.
It is known that Riesz-Frechet representation also holds for quaternionic Hilbert spaces  \cite{GHILONIMORETTIPEROTTI}. Riesz-Frechet representation has also been derived for certain classes of Banach spaces by Giles \cite{GILES} and James \cite{JAMES}.
The traditional proof of the Riesz-Frechet representation uses the orthogonal decomposition theorem, which rests on the parallelogram law  \cite{HALMOS2, PAVONE}. 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 \cite{HALMOS}.
We are fundamentally motivated by the following question.
\begin{question}
What is the non-Archimedean Riesz-Frechet representation?
\end{question}
We begin by recalling the definition of non-Archimedean Banach and p-adic Hilbert spaces.
\begin{definition}\cite{SCHIKHOF}
	Let $\mathbb{K}$ be a field. A map $|\cdot|:\mathbb{K} \to [0, \infty)$ is said to be a \textbf{non-Archimedean valuation} if the following conditions hold:
	\begin{enumerate}[\upshape(i)]
		\item If $\lambda \in \mathbb{K}$ satisfies $|\lambda|=0$, then $\lambda=0$. 
		\item $|\lambda\mu|=|\lambda||\mu|$ for all $\lambda, \mu \in \mathbb{K}$.
		\item (Ultratriangle inequality) $|\lambda+\mu|\leq \max\{\lambda|, |\mu|\}$ for all $\lambda, \mu \in \mathbb{K}$.
	\end{enumerate}
	We say $\mathbb{K}$ is \textbf{non-Archimedean complete valued field} if it is complete w.r.t. the ultrametric 
	\begin{align*}
		d(\lambda, \mu)\coloneqq |\lambda-\mu|, \quad \forall \lambda, \mu \in \mathbb{K}.
	\end{align*}
\end{definition}



\begin{definition}\cite{SCHNEIDER}
	Let $\mathbb{K}$ be a non-Archimedean complete valued field. A vector space $\mathcal{X}$ over $\mathbb{K}$ is said to be \textbf{non-Archimedean Banach space} if there is a map (called as ultra-norm) $\|\cdot\|:\mathcal{X} \to [0, \infty)$ satisfying the following conditions:
	\begin{enumerate}[\upshape(i)]
		\item If $x \in \mathcal{X}$ is such that $\|x\|=0$, then $x=0$.
		\item $\|\lambda x\|=|\lambda|\|x\|$ for all $\lambda \in \mathbb{K}$, for all $x \in \mathcal{X}$.
		\item  (Ultranorm inequality) $\|x+y\|\leq \max\{\|x\|, \|y\|\}$ for all $x, y \in \mathcal{X}$.
		\item $\mathcal{X}$ is complete w.r.t. the ultrametric 
		\begin{align*}
			d(x,y)\coloneqq \|x-y\|, \quad \forall x, y \in \mathcal{X}.
		\end{align*}
	\end{enumerate}
\end{definition}

\begin{definition}\cite{KALISCH} \label{PADICDEF}
	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 \textbf{p-adic Hilbert space} if there is a map (called as \textbf{p-adic inner product}) $\langle \cdot, \cdot \rangle: \mathcal{X} \times \mathcal{X} \to \mathbb{K}$ satisfying the following conditions:
	\begin{enumerate}[\upshape (i)]
		\item If $x \in \mathcal{X}$ is such that $\langle x,y \rangle =0$ for all $y \in \mathcal{X}$, then $x=0$.
		\item $\langle x, y \rangle =\langle y, x \rangle$ for all $x,y \in \mathcal{X}$.
		\item $\langle \alpha x, y \rangle =\alpha \langle x, y \rangle$ for all $\alpha  \in \mathbb{K}$, for all $x,y \in \mathcal{X}$.
		\item $\langle x+y, z \rangle = \langle x,  z \rangle+\langle y,z\rangle$ for all $x,y,z \in \mathcal{X}$.
		\item $|\langle x, y \rangle |\leq \|x\|\|y\|$ for all $x,y \in \mathcal{X}$.
	\end{enumerate}
\end{definition}
Definition \ref{PADICDEF} is more general than the definition of p-adic Hilbert spaces used by Kalisch, which imposes further conditions \cite{KALISCH}. We note that the Definition \ref{PADICDEF} is different than the p-adic Hilbert spaces defined recently by Claussnitzer and Thom \cite{CLAUSSNITZERTHOM} where an exact Riesz-Frechet representation is known \cite{BUSSGARCIAMUKHERJEE}.
We derive two different versions of non-Archimedean Riesz-Frechet representation with different assumptions and different conclusions. 



\section{Non-Archimedean Riesz-Frechet Representation}
In the paper, we use the following notion of Schauder basis for non-Archimedean Banach spaces and orthonormal basis for p-adic Hilbert spaces.
\begin{definition}
	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 \textbf{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 
	\begin{align*}
		x=\sum_{n=1}^{\infty}\alpha _n (x)\tau_n, 
	\end{align*}
	series converges in the ultra-norm on $\mathcal{X}$.
\end{definition}
\begin{definition}\label{PONB}
	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 \textbf{p-adic orthonormal basis} for    $\mathcal{X}$   if  the following conditions hold:
	\begin{enumerate}[\upshape(i)]
		\item $\|\tau_n\|\leq 1$ for all $n \in \mathbb{N}$.
		\item $	\langle \tau_n, \tau_m\rangle =\delta_{n ,m}$ for all $ n, m \in \mathbb{N}$.
	\end{enumerate}
\end{definition}
Note that given a p-adic orthonormal basis $\{\tau_n\}_{n}$ for  $\mathcal{X}$,  condition (ii) in Definition \ref{PONB} says that 
\begin{align*}
	1=|\langle \tau_n, \tau_n \rangle |\leq \|\tau_n\|^2, \quad \forall n \in \mathbb{N}.
\end{align*}
By using condition (i) in Definition \ref{PONB} 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. 
\begin{example}\label{SE}
	Let $\mathbb{K}$ be a non-Archimedean valued field. Define 
	\begin{align*}
		c_0(\mathbb{N}, \mathbb{K})\coloneqq \left\{\{a_n\}_n: a_n \in \mathbb{K}, \forall n \in \mathbb{N}, \lim_{n \to \infty}a_n=0\right\}.
	\end{align*}
	We define 
	\begin{align*}
		\|\{a_n\}_n\|\coloneqq 	\max_{n \in \mathbb{N}}|a_n|, \quad \forall \{a_n\}_n\in c_0(\mathbb{N}, \mathbb{K})
	\end{align*}
	and 
	\begin{align*}
		\langle \{a_n\}_n, \{b_n\}_n\rangle \coloneqq \sum_{n=1}^{\infty}a_nb_n, \quad \forall \{a_n\}_n, \{b_n\}_n \in c_0(\mathbb{N}, \mathbb{K}).
	\end{align*}
	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 $c_0(\mathbb{N}, \mathbb{K})$.
\end{example}
Gram-Schmidt orthonormalization says that every separable Hilbert space has an orthonormal basis \cite{LEONBJORCKGANDER}. Eventhough we know from Example \ref{SE} 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.
\begin{theorem} (\textbf{Non-Archimedean Riesz-Frechet Representation})\label{p1} 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 
	\begin{align*}
		\lim_{n \to \infty}f(\omega_n)\omega_n=0.
	\end{align*}
	Then there is a unique element $\tau \in \mathcal{X}$ satisfying the following conditions:
	\begin{enumerate}[\upshape(i)]
		\item $f(x)=\langle x, \tau \rangle$ for all $x \in \mathcal{X}$.
		\item $\|f\|=\|\tau\|$.
	\end{enumerate}
\end{theorem}
	\begin{proof}
		Since $\{\omega_n\}_{n}$ is orthonormal, 
		\begin{align*}
			\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}.
		\end{align*}
	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 
	\begin{align*}
		\tau\coloneqq \sum_{n=1}^{\infty}f(\omega_n)\omega_n.
	\end{align*}
We then have 
\begin{align*}
	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{align*}
Now 
\begin{align*}
	\|f\|=\sup\left\{\frac{|f(x)|}{\|x\|}: x \in \mathcal{X}, x \neq 0\right\}=\sup\left\{\frac{|\langle x, \tau \rangle|}{\|x\|}: x \in \mathcal{X}, x \neq 0\right\}\leq \|\tau\|.
\end{align*}
 We now observe that 
	\begin{align*}
		\|\tau\|=\left\|\sum_{n=1}^{\infty}f(\omega_n)\omega_n\right\|=\max_{n \in \mathbb{N}}|f(\omega_n)|\leq \|f\|\max_{n \in \mathbb{N}}\|\omega_n\|=\|f\|.
	\end{align*}
Uniqueness easily follows using the definition of p-adic inner product.
	\end{proof}
\begin{example}
	Let $p$ be a prime number. Define 
	\begin{align*}
		f:c_0(\mathbb{N}, \mathbb{Q}_p)\ni \{a_n\}_n\mapsto f( \{a_n\}_n)\coloneqq\sum_{n=1}^{\infty}p^na_n \in \mathbb{Q}_p.
	\end{align*}
Let $\{a_n\}_n \in c_0(\mathbb{N}, \mathbb{Q}_p)$. Then 
\begin{align*}
	|p^na_n|_p=\frac{1}{p^n}|a_n|_p\to 0 \text{ as } n \to \infty.
\end{align*}
Hence
\begin{align*}
	\lim_{n\to \infty}p^na_n=0, \quad \forall \{a_n\}_n \in c_0(\mathbb{N}, \mathbb{Q}_p).
\end{align*}
So the map $f$ is well-defined. Define 
\begin{align*}
	\tau\coloneqq \left\{p^n\right\}_n \in c_0(\mathbb{N}, \mathbb{Q}_p).
\end{align*}
Then 
\begin{align*}
	f(x)=\langle x, \tau \rangle, ~ \forall x \in c_0(\mathbb{N}, \mathbb{Q}_p) \quad \text{and} \quad \|f\|=\|\tau\|=\max_{n \in \mathbb{N}}|p^n|_p=\max_{n \in \mathbb{N}}\left\{\frac{1}{p^n}\right\}=\frac{1}{p}.
\end{align*}
\end{example}
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.
\begin{example}
	Let $p$ be a prime number. Define 
\begin{align*}
	f:c_0(\mathbb{N}, \mathbb{Q}_p)\ni \{a_n\}_n\mapsto f( \{a_n\}_n)\coloneqq\sum_{n=1}^{\infty}na_n \in \mathbb{Q}_p.
\end{align*}
Let $\{a_n\}_n \in c_0(\mathbb{N}, \mathbb{Q}_p)$. Then 
\begin{align*}
	|na_n|_p=|n||a_n|_p\leq |a_n|_p\to 0 \text{ as } n \to \infty.
\end{align*}
Hence
\begin{align*}
	\lim_{n\to \infty}na_n=0, \quad \forall \{a_n\}_n \in c_0(\mathbb{N}, \mathbb{Q}_p).
\end{align*}
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 $\tau\coloneqq\{a_n\}_n \in  c_0(\mathbb{N}, \mathbb{Q}_p)$. Let $\{e_n\}$ be the standard p-adic orthonormal basis for $c_0(\mathbb{N}, \mathbb{Q}_p)$. Now we see that 
\begin{align*}
	n=f(e_n)=\langle e_n, \tau\rangle=a_n, \quad \forall n \in \mathbb{N}.
\end{align*}
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. 
\end{example}
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 
\begin{enumerate}[\upshape(i)]
	\item $x=y+z$.
	\item $\langle y, z \rangle =0$.
	\item $\|x\|=\max\{\|y\|, \|z\|\}$.
\end{enumerate}
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.
\begin{theorem}(\textbf{Non-Archimedean Riesz-Frechet Representation}) \label{p2}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 $\langle \omega, \omega \rangle\neq 0$. 
Then there is a unique element $\tau \in \mathcal{X}$ satisfying the following conditions:
\begin{enumerate}[\upshape(i)]
	\item $f(x)=\langle x, \tau \rangle$ for all $x \in \mathcal{X}$.
	\item 
	\begin{align*}
		\frac{|\langle \tau, \tau \rangle|}{\|\tau\|}\leq \|f\|\leq \|\tau\|.
	\end{align*}
\end{enumerate}
Moreover, 
\begin{align*}
	\|\tau\|=\frac{|f(\omega)|}{|\langle \omega, \omega \rangle|} \|\omega\|\leq \frac{\|f\|\|\omega\|^2}{|\langle \omega, \omega \rangle|}.
\end{align*}
\end{theorem}	
\begin{proof}
	Define 
	\begin{align*}
		\tau\coloneqq \frac{f(\omega)}{\langle \omega, \omega \rangle} \omega.
	\end{align*}
Let $x \in \mathcal{X}$. Then 
\begin{align*}
	f\left(x-\frac{f(x)}{f(\omega)}\omega\right)=f(x)-f(x)=0.
\end{align*}
Hence 
\begin{align*}
	x-\frac{f(x)}{f(\omega)}\omega \in \operatorname{Ker}(f).
\end{align*}
We then have 
\begin{align*}
	0&=\left\langle x-\frac{f(x)}{f(\omega)}\omega, \tau \right\rangle =\left\langle x, \tau \right\rangle-\left\langle \frac{f(x)}{f(\omega)}\omega, \tau \right\rangle\\
	&=\left\langle x, \tau \right\rangle- \frac{f(x)}{f(\omega)}\langle \omega, \tau \rangle=\left\langle x, \tau \right\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{align*}
Note that
\begin{align*}
	|\langle \tau, \tau \rangle|=|f(\tau)|\leq \|f\|\|\tau\|.
	\end{align*}
\end{proof}
We next extend the non-Archimedean Riesz-Frechet representation to non-Archimedean fields  admitting conjugation (involution). We use the following definition of conjugation.
\begin{definition}
	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 \textbf{conjugation} or \textbf{involution} if the following conditions hold:
	\begin{enumerate}[\upshape(i)]
		\item $\sigma(\alpha+\beta)=\sigma(\alpha)+\sigma(\beta)$ for all $\alpha, \beta \in \mathbb{K}$.
		\item  $\sigma(\alpha\beta)=\sigma(\alpha)\sigma(\beta)$ for all $\alpha, \beta \in \mathbb{K}$.
		\item $\sigma(\sigma(\alpha))=\alpha$ for all $\alpha  \in \mathbb{K}$.
		\item $|\sigma(\alpha)|=|\alpha|$ for all $\alpha  \in \mathbb{K}$.
	\end{enumerate}
	In this case, we say that $\mathbb{K}$ is  $\sigma$-conjugated. 
\end{definition}
\begin{definition}
	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 \textbf{p-adic $\sigma$-Hilbert space} if there is a map (called as \textbf{p-adic $\sigma$-inner product}) $\langle \cdot, \cdot \rangle: \mathcal{X} \times \mathcal{X} \to \mathbb{K}$ satisfying the following:
	\begin{enumerate}[\upshape (i)]
		\item If $x \in \mathcal{X}$ is such that $\langle x,y \rangle =0$ for all $y \in \mathcal{X}$, then $x=0$.
	\item $\langle x, y \rangle =\sigma(\langle y, x \rangle)$ for all $x,y \in \mathcal{X}$.
	\item $\langle \alpha x, y \rangle =\alpha \langle x, y \rangle$ for all $\alpha  \in \mathbb{K}$, for all $x,y \in \mathcal{X}$.
	\item $\langle x+y, z \rangle = \langle x,  z \rangle+\langle y,z\rangle$ for all $x,y,z \in \mathcal{X}$.
	\item $|\langle x, y \rangle |\leq \|x\|\|y\|$ for all $x,y \in \mathcal{X}$.
	\end{enumerate}
\end{definition}
We now have p-adic versions of Theorem \ref{p1} and Theorem \ref{p2}.
\begin{theorem} (\textbf{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 
	\begin{align*}
		\lim_{n \to \infty}f(\omega_n)\omega_n=0.
	\end{align*}
	Then there is a unique element $\tau \in \mathcal{X}$ satisfying the following conditions:
	\begin{enumerate}[\upshape(i)]
		\item $f(x)=\langle x, \tau \rangle$ for all $x \in \mathcal{X}$.
		\item $\|f\|=\|\tau\|$.
	\end{enumerate}
\end{theorem}
\begin{proof}
	Define 
	\begin{align*}
		\tau\coloneqq \sum_{n=1}^{\infty}\sigma(f(\omega_n))\omega_n.
	\end{align*}
The other arguments are similar to those used in the proof of Theorem \ref{p1}.
\end{proof}
\begin{theorem}(\textbf{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 $\langle \omega, \omega \rangle\neq 0$. 
	Then there is a unique element $\tau \in \mathcal{X}$ satisfying the following conditions:
	\begin{enumerate}[\upshape(i)]
		\item $f(x)=\langle x, \tau \rangle$ for all $x \in \mathcal{X}$.
		\item 
		\begin{align*}
			\frac{|\langle \tau, \tau \rangle|}{\|\tau\|}\leq \|f\|\leq \|\tau\|.
		\end{align*}
	\end{enumerate}
	Moreover, 
	\begin{align*}
		\|\tau\|=\frac{|f(\omega)|}{|\langle \omega, \omega \rangle|} \|\omega\|\leq \frac{\|f\|\|\omega\|^2}{|\langle \omega, \omega \rangle|}.
	\end{align*}
\end{theorem}	
\begin{proof}
		Define 
	\begin{align*}
		\tau\coloneqq \frac{\sigma(f(\omega))}{\langle \omega, \omega \rangle} \omega.
	\end{align*}
	The other arguments are similar to those used in the proof of Theorem \ref{p2}.
\end{proof}
\section{Conclusions}
\begin{enumerate}
	\item In 1907, Riesz \cite{RIESZ} and Frechet \cite{FRECHET} independently showed that every continuous linear functional on a Hilbert space is completely determined by a single element in that Hilbert space \cite{BIRKHOFFKREYSZIG}.
	\item In 1938, Markov \cite{MARKOV} and in 1941, Kakutani \cite{KAKUTANI} extended the Riesz-Frechet representation to function spaces. 
	\item In 1947, James \cite{JAMES} and  1967, Giles \cite{GILES} extended the Riesz-Frechet representation to certain classes of Banach spaces. 
	\item In this article, we derived two non-Archimedean Riesz representation theorems under certain conditions for p-adic Hilbert spaces. 
\end{enumerate}





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