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.

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.

Preliminaries

Definition 2.1

([5, 6])

A normed linear space E is said to be:

  1. (1)

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

  2. (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. (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. (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 \(\lim_{t\to \infty } \vartheta (t)=\infty \).

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. (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. (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. (1)

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

  2. (2)

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

  3. (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. (4)

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

  5. (5)

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

  6. (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 \(M_{J}^{\perp }= M_{J}^{o} \cap (-M)_{J}^{o} =M_{J}^{o}\). □

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.

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Abbreviations

SCN:

smooth countably normed (space)

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Acknowledgements

We are thankful to the reviewers for their careful reading and valuable comments, which considerably improved this paper.

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The authors were supported financially while writing this paper only from their own personal sources.

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Correspondence to Sarah Tawfeek.

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Tawfeek, S., Faried, N. & El-Sharkawy, H.A. Orthogonality in smooth countably normed spaces. J Inequal Appl 2021, 20 (2021). https://doi.org/10.1186/s13660-020-02531-5

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MSC

  • 46A04

Keywords

  • Countably normed space
  • Normalized duality mapping
  • J-orthogonality
  • Uniformly convex countably normed space
  • Projection theorem in a countably normed space
  • Metric projection
  • Birkhoff orthogonality
  • J-dual cone
  • J-orthogonal complement