Orthogonality in smooth countably normed spaces

Abstract

We generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

1 Introduction

The concept of duality mapping was introduced by Beurling and Livingston [1] in a geometric form. A slightly extended version of the concept was proposed by Asplund [2], who showed how the duality mappings can be characterized via the subdifferentials of convex functions. It is well known that the geometric properties of a Banach space E correspond to the analytic properties of the duality mapping, and it is recognized that if E is smooth, then the duality mapping is single-valued. Park and Rhee [3] defined J-orthogonality in a smooth Banach space using the normalized duality mapping. In this paper, we define the normalized duality mapping on smooth countably normed spaces, generalize the concepts of J-orthogonality and Birkhoff orthogonality in smooth countably normed spaces, and give some basic properties of J-orthogonality in these spaces. Faried and El-Sharkawy [4] defined real uniformly convex complete countably normed spaces and proved that the metric projection on a nonempty convex and closed proper subset of these spaces is well defined. In this paper, we give a relation between metric projection and J-orthogonality and show fundamental links between metric projection and normalized duality mapping in smooth uniformly convex complete countably normed spaces.

2 Preliminaries

Definition 2.1

([5, 6])

A normed linear space E is said to be:

• (1) Strictly convex if $\|\frac{x+y}{2}\|<1$ for all $x ,y \in E$ with $\|x\|=\|y\|=1$ and $x \neq y$;

• (2) Uniformly convex if for any $\varepsilon \in (0,2]$, there exists $\delta =\delta (\varepsilon )> 0$ such that if $x,y \in E$ with $\|x\|=1$, $\|y\|=1$, and $\|x-y\| \geq \varepsilon$, then $\|\frac{x+y}{2}\| \leq 1-\delta$;

• (3) Smooth if $\lim_{t\to 0} \frac{\|x+ty\|-\|x\|}{t}$ exists for all $x,y \in S(E)$, where $S(E)$ is the unit sphere of E;

• (4) Uniformly smooth if for every $\varepsilon > 0$, there exists $\delta > 0$ such that for all $x,y\in E$ with $\|x\|=1$ and $\|y\|\leq \delta$, we have $\|x+y\|+\|x-y\|< 2+\varepsilon \|y\|$.

Definition 2.2

(Metric projection [6])

Let E be a real uniformly convex Banach space, and let K be a nonempty proper subset of E. The operator $P_{K} : E\rightarrow K$ is called a metric projection operator if it assigns to each $x\in E$ its nearest point $\bar{x} \in K$, that is, the solution of the minimization problem

$$ P_{K} x=\bar{x} : \Vert x-\bar{x} \Vert =\inf _{y\in K} \Vert x-y \Vert . $$

Definition 2.3

(The normalized duality mapping [7, 8])

Let E be a real Banach space with norm $\| \cdot \|$, and let $E^{*}$ be the dual space of E, and let $\langle \cdot ,\cdot \rangle$ be the duality pairing. The normalized duality mapping J from E to $2^{E^{*}}$ is defined by

$$ Jx=\bigl\{ x^{*} \in E^{*} : \bigl\langle x, x^{*}\bigr\rangle = \Vert x \Vert ^{2}= \bigl\Vert x^{*} \bigr\Vert ^{2}\bigr\} . $$

The Hahn–Banach theorem guarantees that $Jx \neq \emptyset$ for every $x\in E$. It is well known that if E is a smooth Banach space, then the normalized duality mapping is single-valued. In [8], we got the following example of the normalized duality mapping J in the uniformly convex and uniformly smooth Banach space $\ell ^{p}$ with $p\in (1,\infty )$: $Jx:=\|x\|^{2-p}_{\ell ^{p}}\{x_{1}|x_{1}|^{p-2}, x_{2}|x_{2}|^{p-2},\ldots \} \in \ell ^{q} = {\ell ^{p}}^{*}$ for $x=\{x_{1},x_{2},\ldots\} \in \ell ^{p}$, where $\frac{1}{p}+\frac{1}{q}=1$.

Proposition 2.4

([9])

Let E be a smooth Banach space, let $E^{*}$ be the dual space of E, and let J be the normalized duality mapping from E to $2^{E^{*}}$. Then J is a continuous operator in E, and $J(\beta x)=\beta J(x)$ for all $\beta \in \mathbb{R}$.

Definition 2.5

(Lyapunov functional [7, 8])

Let E be a smooth Banach space, and let $E^{*}$ be the dual space of E. The Lyapunov functional $\varphi : E\times E \rightarrow \mathbb{R}$ is defined by

$$ \varphi (y,x)= \Vert y \Vert ^{2} -2 \langle y,Jx \rangle + \Vert x \Vert ^{2} $$

for all $x,y\in E$, where J is the normalized duality mapping from E to $2^{E^{*}}$.

Definition 2.6

(Compatible norms [10, 11])

Two norms in a linear space E are said to be compatible if every Cauchy sequence $\{x_{n}\}$ in E with respect to both norms that converges to a limit $x\in E$ with respect to one of them also converges to the same limit x with respect to the other norm.

Definition 2.7

(Countably normed space [10, 11])

A linear space E equipped with a countable family of pairwise compatible norms $\{\|\cdot \|_{n}, n \in \mathbb{N}\}$ is said to be a countably normed space. An example of a countably normed space is the space $\ell ^{p+0} :=\bigcap_{n} \ell ^{p_{n}}$ ($1< p < \infty$) for any choice of a decreasing sequence ${p_{n}}$ converging to p.

Remark 2.8

([11])

For a countably normed space $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$, let $E_{n}$ be the completion of E with respect to the norm $\|\cdot \|_{n}$. We may assume that $\|\cdot \|_{1} \leq \|\cdot \|_{2} \leq \|\cdot \|_{3} \leq \cdots$ in any countably normed space; we also have $E\subset \cdots \subset E_{n+1} \subset E_{n} \subset \cdots \subset E_{1}$.

Proposition 2.9

([10])

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a countably normed space. Then E is complete if and only if $E=\bigcap_{n \in \mathbb{N}} E_{n}$. Each Banach space $E_{n}$ has a dual $E_{n}^{*}$, which is a Banach space, and the dual of the countably normed space E is given by $E^{*}=\bigcup_{n \in \mathbb{N}} E_{n} ^{*}$. We have the following inclusions:

$$ {E_{1}^{*}} \subset \cdots \subset {E_{n}^{*}} \subset {E_{n+1}^{*}} \subset \cdots \subset {E^{*}}. $$

Moreover, for $f\in E_{n}^{*}$, we have $\|f\|_{n}\geq \|f\|_{n+1}$ for all $n \in \mathbb{N}$.

Definition 2.10

(Uniformly convex countably normed space [4])

A countably normed space $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ is said to be uniformly convex if $(E_{n}, \| \cdot \|_{n})$ is uniformly convex for all $n \in \mathbb{N}$.

Theorem 2.11

([4])

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a real uniformly convex complete countably normed space, and let K be a nonempty convex proper subset of E such that K is closed in each $E_{n}$. Then there exists a unique $\bar{x} \in K$ such that $\|x-\bar{x}\|_{n} =\inf_{y \in K} \|x - y\|_{n}$ for all $n \in \mathbb{N}$, and the metric projection $P : E\rightarrow K$ is defined by $P(x)=\bar{x}$.

Definition 2.12

(J-orthogonality [3])

Let E be a smooth Banach space. Two elements $x,y \in E$ are said to be J-orthogonal, written “x is J-orthogonal to y” or $x\perp ^{J} y$, if $\langle y, Jx\rangle =0$.

Definition 2.13

(Gauge function [8])

A gauge function is a continuous strictly increasing function $\vartheta : \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $\vartheta (0)=0$ and $\lim_{t\to \infty } \vartheta (t)=\infty$.

3 Main results

Now we introduce the concept of the normalized duality mapping in smooth countably normed (SCN) spaces.

Definition 3.1

(The normalized duality mapping in SCN spaces)

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a smooth countably normed space such that $E_{n}$ is the completion of E in $\|\cdot \|_{n}$ and $(E_{n} , \|\cdot \|_{n})$ is a smooth Banach space for all $n \in \mathbb{N}$, so that there exists a single-valued normalized duality mapping $J_{n} : E_{n}\rightarrow E_{n}^{*}$ with respect to $\|\cdot \|_{n}$ for all $n \in \mathbb{N}$. Without being confused, we understand that $\|J_{n}x\|_{n}$ is the $E_{n}^{*}$-norm and $\|x\|_{n}$ is the $E_{n}$-norm, for all $n \in \mathbb{N}$.

We define the following multivalued mapping $J : E\rightarrow 2^{E^{*}}$ to be the normalized duality mapping of a smooth countably normed space: $J(x)= \{J_{n}x\}_{n=1}^{\infty } \subseteq E^{*}=\bigcup_{n \in \mathbb{N}} E_{n} ^{*}$, $\|J_{n}x\|_{n}=\|x\|_{n}$, $\langle J_{n}x,x\rangle =\|x \|_{n}^{2}$ for $n \in \mathbb{N}$.

Remark 3.2

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a smooth countably normed space. The sequence of norms is increasing in E, and from the definition of normalized duality mappings $J_{n}$ for each $E_{n}$ with respect to $\|\cdot \|_{n}$ we have

$$ \bigl( \Vert x \Vert _{1}= \Vert J_{1}x \Vert _{1}\bigr) \leq \bigl( \Vert x \Vert _{2}= \Vert J_{2}x \Vert _{2}\bigr) \leq \cdots \leq \bigl( \Vert x \Vert _{n}= \Vert J_{n}x \Vert _{n}\bigr) \leq \cdots , $$

and thus $\langle J_{1}x,x\rangle \leq \langle J_{2}x,x\rangle \leq \cdots \leq \langle J_{n}x,x\rangle \leq \ldots$ , and using the properties of countably normed spaces, we have $\|J_{i}x\|_{n}\geq \|J_{i}x\|_{n+1}$ for all i and n.

Remark 3.3

The multivalued normalized duality mapping of a smooth countably normed space cannot be a single-valued mapping, unlike the case of a smooth Banach space. Indeed, if it were a single-valued mapping, then it would be the same single-valued normalized duality mapping for each $E_{n}$ with respect to $\|\cdot \|_{n}$, which would imply that $\langle Jx,x\rangle =\|x\|_{n}^{2}$ for all n. Then we would get $\|x\|_{1}=\|x\|_{2}=\cdots =\|x\|_{n}=\cdots$ , which would mean that we are back to a normed vector space, and this ruins the construction of the countably normed space.

Proposition 3.4

If $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ is a smooth countably normed space, then $J_{m}|_{E_{n}}$ is the single-valued normalized duality mapping of $E_{n}$ with respect to $\|\cdot \|_{n}$ for all $m=1,2,\ldots ,n-1$ and $n\geq 2$.

Proof

Let $J_{n-1}$ be the normalized duality mapping of $E_{n-1}$ with respect to $\|\cdot \|_{n-1}$. We have $J_{n-1} : E_{n-1} \rightarrow E_{n}^{*}$, $E_{n-1}^{*} \subseteq E_{n}^{*}$, $E_{n} \subseteq {E_{n-1}}$, so $J_{n-1}|_{E_{n}} : E_{n}\rightarrow E_{n}^{*}$ and $\|J_{n-1}|_{E_{n}} x\|_{n}=\|x\|_{n-1}$, $\langle J_{n-1} |_{E_{n}}x,x \rangle =\|x\|_{n-1}^{2}$ for all $x \in E_{n}\subseteq E_{n-1}$. So $J_{n-1}|_{E_{n}}$ is the single-valued normalized duality mapping of $E_{n}$ with respect to $\|\cdot \|_{n-1}$. The same holds for all $m=1,2,\ldots ,n-1$, and hence $J_{m}|_{E_{n}}$ is the single-valued normalized duality mapping of $E_{n}$ with respect to $\|\cdot \|_{n}$ for all $n\geq 2$. □

Corollary 3.5

If $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ is a smooth countably normed space, then $E_{n}$ is a smooth Banach space with respect to $\|\cdot \|_{m}$, $m=1,2,\ldots , n-1$, $n\geq 2$.

Proof

Since $J_{m}|_{E_{n}}$ is the single-valued normalized duality mapping of $E_{n}$ with respect to $\|\cdot \|_{n}$ for all $m=1,2,\ldots ,n-1$, then $E_{n}$ is a smooth Banach space with respect to $\|\cdot \|_{m}$ for all $n\geq 2$. □

Proposition 3.6

Let E be a smooth countably normed space, let $E^{*}$ be its dual space, and let $J_{n}$ be the normalized duality mapping of $E_{n}$ with respect to $\|\cdot \|_{n}$ relative to the gauge function $\vartheta _{n}$, where $\vartheta _{n} (\|x\|_{n})=\|x\|_{n} = \|J_{n}x\|_{n}$. Define $\psi _{n}(r)=\int _{0}^{r} \vartheta _{n}(\sigma ) \, d \sigma$. Then $\psi _{n}(\|y\|_{n})-\psi _{n}(\|x\|_{n})\geq \langle J_{n}x,y-x \rangle$ for all $y\in E$ and $n \in \mathbb{N}$.

Proof

We have

$$ \psi _{n}\bigl(\|y\|_{n}\bigr)-\psi _{n}\bigl(\|x\|_{n}\bigr)= \int _{\|x\|_{n}}^{ \|y\|_{n}} \vartheta _{n}(t) \,dt\geq \vartheta _{n}\bigl(\|x\|_{n}\bigr)\bigl(\|y\|_{n}- \|x\|_{n}\bigr) ,\quad \forall n, $$

that is, $\psi _{n}(\|y\|_{n})-\psi _{n}(\|x\|_{n})=\vartheta _{n}(\|x\|_{n}) \|y\|_{n}- \langle J_{n}x,x \rangle \geq \langle J_{n}x,y-x \rangle$ for all $y\in E$ and $n \in \mathbb{N}$. □

Proposition 3.7

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a real smooth uniformly convex complete countably normed space, and let K be a nonempty proper convex subset of E such that K is closed in each $E_{n}$. Then $\bar{x}=P_{K}(x)$ is the metric projection of an arbitrary element $x \in E$ if and only if $\langle J(x- \bar{x}), \bar{x}-y \rangle \geq 0$ for all $y \in K$, where J is the normalized duality mapping on E.

Proof

“⇒” By the definition of the metric projection and the convexity of K we have

$$ \|x- \bar{x}\|_{n} \leq \bigl\| x-\bigl(\mu y+(1-\mu )\bar{x}\bigr)\bigr\| _{n} ,\quad \forall y \in K , \mu \in [0,1] ,\forall n. $$

(*)

Consider $\psi _{n}(r)= \int _{0}^{r} \vartheta _{n}(\sigma ) \, d \sigma$. If $J_{n}$ is the normalized duality mapping relative to the gauge function $\vartheta _{n}$ with respect to $\|\cdot \|_{n}$, then (*) is equivalent to

$$ \psi _{n}\bigl(\|x- \bar{x}\|_{n}\bigr) \leq \psi _{n} \bigl( \bigl\| x - \bigl[ \mu y +(1- \mu ) \bar{x} \bigr] \bigr\| _{n} \bigr). $$

(**)

By Proposition 3.6 and (**) we get

$$ 0 \geq \psi _{n}\bigl( \Vert x- \bar{x} \Vert _{n} \bigr)-\psi _{n}\bigl( \bigl\Vert x-\bigl(\mu y+ (1-\mu ) \bar{x} \bigr) \bigr\Vert _{n}\bigr)) \geq \bigl\langle J_{n} \bigl(x- \bar{x} - \mu (y-\bar{x})\bigr), \mu (y- \bar{x}) \bigr\rangle . $$

As μ tends to 0, we get $\langle J_{n}(x- \bar{x}),y-\bar{x} \rangle \leq 0$ for all $y \in K$ and n, that is, $\langle J_{n}(x- \bar{x}), \bar{x}-y \rangle \geq 0$ for all $y \in K$ and n.

“⇐” If $\langle J_{n}(x- \bar{x}), \bar{x} -y \rangle \geq 0$ for all $y \in K$ and n, then using Proposition 3.6, we get

$$ \psi _{n}\bigl( \Vert x-y \Vert _{n}\bigr)-\psi _{n}\bigl( \Vert x- \bar{x} \Vert _{n}\bigr) \geq \bigl\langle J_{n}(x- \bar{x}), \bar{x}-y\bigr\rangle \geq 0. $$

Thus $\|x- \bar{x}\|_{n} \leq \|x-y\|_{n}$ for all $y \in K$ and n, and so $\bar{x}= P_{K}(x)$. □

Theorem 3.8

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a real smooth uniformly convex complete countably normed space, and let K be a nonempty proper convex subset of E such that K is closed in each $E_{n}$.

Then $\bar{x}=P_{K}(x)$ is the metric projection of an arbitrary element $x \in E$ if and only if $\langle J_{n}(x- \bar{x}),x-y\rangle \geq \|x- \bar{x}\|_{n}^{2}$ for all $y \in K$ and n.

Proof

“⇒” By Proposition 3.6 we have $\langle J_{n}(x- \bar{x}), \bar{x}-y\rangle \geq 0$ for all $y \in K$ and n. Besides,

$$\begin{aligned} \bigl\langle J_{n}(x- \bar{x}), \bar{x}-y\bigr\rangle =& J_{n}(x- \bar{x}) (\bar{x}-y) \\ =&J_{n}(x- \bar{x}) (\bar{x}-x)+J_{n}(x-\bar{x}) (x-y) \\ =&- \Vert x-\bar{x} \Vert _{n}^{2}+J_{n}(x- \bar{x}) (x-y), \end{aligned}$$

and therefore $\langle J_{n}(x-\bar{x}),x-y\rangle \geq \|x-\bar{x}\|_{n}^{2}$ for all $y \in K$ and n.

“⇐” If $\|x-\bar{x}\|_{n}=0$, then we are done. So, let us assume that $\|x-\bar{x}\|_{n} \neq 0$. Then

$$\begin{aligned} \Vert x-\bar{x} \Vert _{n} \leq & \frac{1}{ \Vert x-\bar{x} \Vert _{n}} \bigl\langle J_{n}(x- \bar{x}),x-y \bigr\rangle \\ \leq & \frac{1}{ \Vert x-\bar{x} \Vert _{n}} \bigl\Vert J_{n}(x-\bar{x}) \bigr\Vert _{n} \Vert x-y \Vert _{n} \\ =& \Vert x-y \Vert _{n},\quad \forall y \in K, \forall n, \end{aligned}$$

that is, $\bar{x}= P_{K}(x)$. □

Definition 3.9

(J-orthogonality in smooth countably normed spaces)

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a smooth countably normed space. We say that an element $x\in E$ is J-orthogonal to an element $y\in E$ and write $x \perp ^{J} y$ if $\langle y, J_{n}x \rangle =0$ for all n, that is, $\langle y,Jx\rangle =0$, where J is the normalized duality mapping of E.

Definition 3.10

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a smooth countably normed space, and let $x_{1}, x_{2}, \ldots , x_{n} \in E\setminus \{0\}$. Then the set $\{x_{1},x_{2},\ldots , x_{n}\}$ is called a J-orthogonal set if $x_{i}\perp x_{j}$ for all $i, j \in \{1,2,\ldots , n\}$ with $i\neq j$.

Definition 3.11

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a smooth countably normed space. We say that an element $x\in E$ is orthogonal to an element $y\in E$ in the Birkhoff sense if $\|x+\alpha y\|_{i}^{2} \geq \|x\|_{i}^{2}$ for all $i=1,2,\ldots , n, \ldots$ and $\alpha \in \mathbb{R}$.

Proposition 3.12

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a smooth countably normed space, and let $x_{1}, x_{2}, \ldots , x_{n} \in E\setminus \{0\}$. Then:

• (1) If $\{x_{1},x_{2},\ldots , x_{n}\}$ is a J-orthogonal set, then $x_{1},x_{2},\ldots , x_{n}$ are linearly independent;

• (2) Let $x,y \in E$. Then $x\perp ^{J} y$ if and only if $x\perp y$ in the Birkhoff sense.

Proof

(1) Let $\alpha _{1} x_{1}+\alpha _{2} x_{2}+\cdots +\alpha _{n} x_{n}=0$ for some scalars $\alpha _{1} , \alpha _{2} , \ldots , \alpha _{n} \in \mathbb{R}$.

For all $m \in \{1, \ldots , n\}$ and i, we have:

$$\begin{aligned} \langle \alpha _{1} x_{1}+\cdots +\alpha _{n} x_{n}, J_{i}x_{m} \rangle =& \alpha _{1} \langle x_{1},J_{i}x_{m} \rangle +\cdots + \alpha _{n} \langle x_{n}, J_{i}x_{m} \rangle \\ =&\alpha _{m} \Vert x_{m} \Vert _{i}^{2} \\ =&0, \end{aligned}$$

and so $\alpha _{m}=0$ for all m. Thus $x_{1},x_{2},\ldots , x_{n}$ are linearly independent.

(2) If $x\perp ^{J} y$, then $\langle y,J_{i}x\rangle =0$ for all i. Besides, using the Lyapunov functional, we have

$$\begin{aligned} \varphi _{i}(x+\alpha y,x) =& \Vert x+\alpha y \Vert _{i}^{2}-2 \langle x+ \alpha y, J_{i}x \rangle + \Vert x \Vert _{i}^{2} ,\quad \forall i \\ =& \Vert x+\alpha y \Vert _{i}^{2}- \Vert x \Vert _{i}^{2}-2\alpha \langle y, J_{i}x \rangle \\ \geq & 0,\quad \forall i, \forall \alpha \in \mathbb{R}. \end{aligned}$$

Thus $\|x+\alpha y\|_{i}^{2} \geq \|x\|_{i}^{2}$ for all i and $\alpha \in \mathbb{R}$. Hence $x\perp y$ in the Birkhoff sense.

On the other hand, let $x\perp y$ in the Birkhoff sense, that is, $\|x+\alpha y\|_{i}^{2} \geq \|x\|_{i}^{2}$ for all i and $\alpha \in \mathbb{R}$. If $\langle y, J_{i} x\rangle \neq 0$ for some i, then by taking $\alpha _{0} = \frac{\|x+\alpha y\|_{i}^{2}-\|x\|_{i}^{2}}{\langle y, J_{i}x\rangle }$ we get that the Lyapunov functional $\varphi _{i} (x+\alpha _{0} y,x) < 0$. This contradicts that $\varphi _{i}(x,y) > 0$ for all i. □

Proposition 3.13

Let $\{x_{1},x_{2},\ldots , x_{n}\}$ be a J-orthogonal set in a smooth countably normed space E with dual space $E^{*}$. The set $\{J_{i}x_{1}, \ldots , J_{i}x_{n}\}$ is linearly independent in the dual space $E^{*}$ for all i.

Proof

If $\alpha _{1} J_{i}x_{1}+ \cdots +\alpha _{n} J_{i}x_{n}=0$ for some scalars $\alpha _{1} , \ldots , \alpha _{n} \in \mathbb{R}$, then for each $m \in \{1,2,\ldots ,n\}$, we get $\langle x_{m},\alpha _{1} J_{i}x_{1}+\cdots + \alpha _{n} J_{i}x_{n} \rangle = \alpha _{m} \|x\|_{i}^{2} = 0$ for all i. Hence $\alpha _{m} =0$ for all m. Thus, for all i, the set $\{J_{i}x_{1}, \ldots , J_{i}x_{n}\}$ is linearly independent in the dual space $E^{*}$. □

The following theorem gives a relation between metric projection and orthogonality in real uniformly convex complete countably normed spaces.

Theorem 3.14

Let $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ be a real smooth uniformly convex complete countably normed space, and let M be a nonempty proper subspace of E such that M is closed in each $E_{i}$. Then

$$ \forall x\in E\setminus M, \exists ! \bar{x} \in M \textit{:} \quad \Vert x-\bar{x} \Vert _{i}= \inf_{y\in M} \Vert x-y \Vert _{i} $$

for all i if and only if $x-\bar{x} \bot ^{J} M$.

Proof

Assume that

$$ \forall x\in E\setminus M, \exists ! \bar{x} \in M \mbox{:}\quad \Vert x-\bar{x} \Vert _{i}= \inf_{y\in M} \Vert x-y \Vert _{i} ,\quad \forall i. $$

If $z\in M$, then $\bar{x}-\alpha z \in M$ for all $\alpha \in \mathbb{R}$, and $\|x-\bar{x}\|_{i} \leq \|x-(\bar{x}-\alpha z)\|_{i}=\|(x-\bar{x})+ \alpha z\|_{i}$ for all i. Therefore $x-\bar{x}$ is orthogonal to M in the Birkhoff sense. Consequently, $x-\bar{x}\perp ^{J} M$.

On the other hand, if $x-\bar{x}\perp ^{J} M$, then $x-\bar{x}$ is orthogonal to M in the Birkhoff sense, that is, $\|x-\bar{x}\|_{i} \leq \|x-\bar{x}+\alpha y\|_{i}$ for all $\alpha \in \mathbb{R}$, $y\in M$, and i.

Since $\bar{x}-y \in M$, for all $y\in M$ and i, we get

$$ \Vert x-\bar{x} \Vert _{i}\leq \bigl\Vert x- \bar{x}+\alpha ( \bar{x}-y) \bigr\Vert _{i} $$

for all $\alpha \in \mathbb{R}$.

Taking $\alpha =1$, we get $\|x-\bar{x}\|_{i} \leq \|x-y\|_{i}$ for all $y\in M$ and i. Thus $\|x-\bar{x}\|_{i}=\inf_{y \in M} \|x-y\|_{i}$ for all i. □

Example 3.15

$\ell _{2+0} := \bigcap_{n \in \mathbb{N}} \ell _{2+ \frac{1}{n}}$ is a uniformly convex uniformly smooth complete countably normed space with the norms

$$ \Vert \cdot \Vert _{3} \leq \Vert \cdot \Vert _{2.5}\leq \cdots \leq \Vert \cdot \Vert _{2+ \frac{1}{n}}\leq \cdots $$

for each $x=\{x_{i}\}\in \ell _{2+0}$, and

$$ J_{n}(x)= \Vert x \Vert _{2+\frac{1}{n}}^{-\frac{1}{n}} \bigl\{ x_{i} \vert x_{i} \vert ^{ \frac{1}{n}} \bigr\} \in \ell _{\frac{2n+1}{n+1}} ,\quad \forall n. $$

Consider the closed subspace M of $\ell _{2+0}$ generated by $\{1,0,0,0,\ldots \}$. Using the previous theorem, we get

$$\begin{aligned}& P_{M}(x)= \bar{x}=\{{\bar{x}}_{1},0,0,\ldots\} \\& \quad \Leftrightarrow\quad \bigl\langle \{t,0,0,\ldots \},J_{n}(x-\bar{x})\bigr\rangle =\{0,0,\ldots\} ,\quad \forall t \in \mathbb{R}, \forall n \\& \quad \Leftrightarrow\quad \bigl\langle \{t,0,0,\ldots\},J_{n}\{x_{1}-{ \bar{x}}_{1}, x_{2},x_{3},\ldots,x_{n},\ldots \}\bigr\rangle =\{0,0,\ldots\} \\& \quad \Leftrightarrow\quad \bigl\langle \{t,0,0,\ldots\}, \Vert x-\bar{x} \Vert _{2+\frac{1}{n}}^{- \frac{1}{n}} \bigl\{ \vert x_{1}-{\bar{x}}_{1} \vert ^{-\frac{1}{n}}(x_{1}- {\bar{x}}_{1}),\ldots,x_{i} \vert x_{i} \vert ^{ \frac{1}{n}},\ldots\bigr\} \bigr\rangle = \{0,0,\ldots\} \\& \quad \Leftrightarrow\quad \Vert x-\bar{x} \Vert _{2+\frac{1}{n}}^{-\frac{1}{n}} \vert x_{1}-{ \bar{x}}_{1} \vert ^{-\frac{1}{n}}(x_{1}- {\bar{x}}_{1})t=0 ,\quad\forall t \in \mathbb{R} , \forall n \\& \quad \Leftrightarrow\quad {\bar{x}}_{1}=x_{1},\qquad P_{M}(x)= \bar{x}=\{x_{1},0,0,\ldots \}. \end{aligned}$$

Definition 3.16

The J-dual cone of a nonempty subset S of a smooth countably normed space $(E, \{\|\cdot \|_{n}, n \in \mathbb{N}\})$ is the set

$$ S_{J}^{o}=\bigl\{ x\in E : \langle y,J_{i}x\rangle \leq 0 , \forall y \in S, \forall i\bigr\} . $$

In addition, the J-orthogonal complement of S is the set

$$ S_{J}^{\perp }=S_{J}^{o}\cap (-S)_{J}^{o}=\bigl\{ x\in E : \langle y,J_{i}x \rangle = 0 ,\forall y\in S, \forall i\bigr\} . $$

Theorem 3.17

Let S be a nonempty subset of a smooth countably normed space $(E, \{{\|\cdot \|}_{n}, n \in \mathbb{N}\})$. Then:

• (1) $S_{J}^{o}$ and $S_{J}^{\perp }$ are closed cones;

• (2) $S_{J}^{o}=(\bar{S})_{J}^{o}$ and $S_{J}^{\perp }= (\bar{S})_{J}^{\perp }$;

• (3) $S_{J}^{o}=[\operatorname{conv}(S)]_{J}^{o}={\overline{[\operatorname{conv}(S)]}}_{J}^{o}$ and $S_{J}^{\perp }=[\operatorname{span}(S)]_{J}^{\perp }=\overline{[\operatorname{span}(S)]}_{J}^{\perp }$, where $\operatorname{conv}(S)$ is the convex hull of S, and $\operatorname{span}(S)$ is the subspace generated by S;

• (4) $\bar{S}\subset (S_{J}^{o})^{o}$ and $\bar{S}\subset (S_{J}^{\perp })^{\perp }$;

• (5) If C is a cone, then $(C-y)_{J}^{o}=C_{J}^{o}\cap y_{J}^{\perp }$ for all $y\in C$;

• (6) If M is a subspace, then $M_{J}^{o}=M_{J}^{\perp }$.

Proof

(1) If $x_{n}\in S_{J}^{o}$ and $x_{n}\rightarrow x$, then for all $y \in S$, $\langle y,J_{i}x \rangle =\lim_{n \to \infty } \langle y,J_{i}x_{n} \rangle \leq 0$ ∀i implies that $x \in S_{J}^{o}$, and thus $S_{J}^{o}$ is closed. If $x \in S_{J}^{o}$ and $\alpha \geq 0$, then for all $y \in S$ and i, we get

$$ \bigl\langle y,J_{i}(\alpha x) \bigr\rangle = \langle y,\alpha J_{i}x \rangle = \alpha \langle y,J_{i}x\rangle \leq 0. $$

Hence $\alpha x\in S_{J}^{o}$, and thus $S_{J}^{o}$ is a cone. Since $S_{J}^{\perp }= S_{J}^{o} \cap (-S)_{J}^{o}$, $S_{J}^{\perp }$ is a closed cone.

(2) Since $S \subseteq \bar{S}$, we have $(\bar{S})_{J}^{o}\subseteq S_{J}^{o}$. If $x\in S_{J}^{o}$ and $y\in \bar{S}$, choose $y_{n} \in S$ such that $y_{n}\rightarrow y$. Then $\langle y,J_{i}x\rangle = \lim_{n \to \infty } \langle y_{n},J_{i}x \rangle \leq 0$ for all i implies $x \in (\bar{S})_{J}^{o}$. Thus $S_{J}^{o} = (\bar{S})_{J}^{o}$. Moreover, $S_{J}^{\perp }= (\bar{S})_{J}^{\perp }$.

(3) Since $S\subseteq \operatorname{conv}(S)$, $[\operatorname{conv}(S)]_{J}^{o} \subseteq S_{J}^{o}$. Let $x \in S_{J}^{o}$ and $y \in \operatorname{conv}(S)$. By the definition of $\operatorname{conv}(S)$, $y= \sum_{m=1}^{n} \rho _{m} y_{m}$ for some $y_{i} \in S$ and $\rho _{i} \geq 0$ with $\sum_{m=1}^{n} \rho _{m} =1$, $i=1,2, \ldots ,n$.

Then $\langle y,J_{i}x\rangle = \sum_{m=1}^{n} \rho _{m} \langle y_{m},J_{i}x\rangle \leq 0$ for all i implies $x \in [\operatorname{conv}(S)]_{J}^{o}$, so $S_{J}^{o} \subseteq [\operatorname{conv}(S)]_{J}^{o}$. Thus $S_{J}^{o}=[\operatorname{conv}(S)]_{J}^{o}$. Moreover, $S_{J}^{\perp }= [\operatorname{span}(S)]_{J}^{\perp }= \overline{[\operatorname{span}(S)]}_{J}^{\perp }$.

(4) If $x \in S$, then for all $y \in S_{J}^{o}$, $\langle x,J_{i}y\rangle \leq 0$ for all i. Hence $x\in (S_{J}^{o})^{o}$. Thus $S\subseteq (S_{J}^{o})^{o}$. Since $(S_{J}^{o})^{o}$ is closed, $\bar{S} \subseteq (S_{J}^{o})^{o}$.

(5) Now $x \in (C-y)_{J}^{o}$ if and only if $\langle c-y,J_{i}x\rangle \leq 0$ for all i and $c\in C$. Let $x \in (C-y)_{J}^{o}$. Taking $c=0$ and $c=2y$, we have $\langle y,J_{i}x\rangle =0$, and $\langle c,J_{i}x\rangle \leq 0$ for all i and $c\in C$. Thus $x \in {C_{J}^{o}\cap y_{J}^{\perp }}$. Moreover, if $x \in {C_{J}^{o}\cap y_{J}^{\perp }}$, then $\langle c,J_{i}x\rangle \leq 0$ and $\langle y,J_{i}x\rangle = 0$ for all i and $c \in C$. Thus $x\in (C-y)_{J}^{o}$. Therefore $(C-y)_{J}^{o}= C_{J}^{o} \cap y_{J}^{\perp }$ for all $y \in C$.

(6) If M is a subspace of E, then $-M=M$ implies $M_{J}^{\perp }= M_{J}^{o} \cap (-M)_{J}^{o} =M_{J}^{o}$. □

4 Conclusion

In this paper, we defined J-orthogonality and Birkhoff orthogonality in smooth countably normed spaces and showed that these two types of orthogonality coincide in these spaces. Besides, we proved some basic properties of J-orthogonality in smooth countably normed spaces and gave a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we gave fundamental links between J-orthogonality and metric projection in smooth uniformly convex complete countably normed spaces. In addition, we defined the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and proved some basic results about the J-dual cone and J-orthogonal complement of S.