View of A New Notion of Inner Product in A Subspace of n-Normed Spaces

Vol. 29, No. 03 (2023), pp. 372–381.

A NEW NOTION OF INNER PRODUCT IN A SUBSPACE OF n-NORMED SPACE

Muh. Nur

¹

, and Mochammad Idris

²

1Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Makassar 90245, Indonesia,

muhammadnur@unhas.ac.id

2Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Lambung Mangkurat, Banjarbaru 70714,Indonesia,

moch.idris@ulm.ac.id

Abstract.Given ann-normed spaceXforn≥2, we investigate the completeness ofY (as a subspace ofX) with respect to a new norm that correspond to this new inner product onY. Next, we introduce the angle on a subspace Y ofn-normed spaceX.

Key words and Phrases:n-norm, completness, inner product, angles.

1. INTRODUCTION

An inner product is an important functional in mathematical analysis. On this functional in vector space, we can introduce the orthogonality, the norm, the angle between two subspaces, then-norm, and then-inner product (see [8, 10, 16, 19]). Now letX be a real vector space. We recall that an inner product onX is a mapping ⟨·,·⟩:X²→Rsuch that satisfying

(1) ⟨x, x⟩ ≥0 for allx∈X;x= 0 if and only if⟨x, x⟩= 0;

(2) ⟨x, y⟩=⟨y, x⟩for allx, y∈X;

(3) ⟨γx, y⟩=γ⟨x, y⟩for allx∈X and for any scalarsγ∈R; (4) ⟨x+x^′, y⟩=⟨x, y⟩+⟨x^′, y⟩, for allx, x^′, y∈X .

A pair (X,⟨·,·⟩) is called an inner product space. For instance onRⁿ, we introduce

⟨x, y⟩_Rn :=

n

X

j=1

x_jy_j (1)

for everyx, y∈Rⁿ. Of course, the functional⟨·,·⟩_Rn satisfies 1-4. We can measure

”the length” of x ∈ Rⁿ using ∥x∥_Rn = p

⟨x, x⟩

Rⁿ. It is called a norm of x. In

2020 Mathematics Subject Classification: 46B20, 46C05 46A45, 46B45.

Received: 09-05-2023, accepted: 02-12-2023.

372

general, we now recall a norm on X and explain its properties. It is a function

∥ · ∥:X →Rwhich satifies

(1) ∥x∥ ≥0, for allx∈X;∥x∥= 0 if and only ifx= 0∈X;

(2) ∥γx∥=|γ| ∥x∥, for allx∈X and for all scalarγ∈R; (3) ∥x+x^′∥ ≤ ∥x∥+∥x^′∥ for allx, x^′ ∈X.

We call that a pair of (X,∥ · ∥) is a normed space.

A normed space is the inner product space if it satisfies the parallelogram law (see [16]). There have been many researchers’ efforts in formulating ”the inner product” in normed space. See [15, 20] for a new inner products with a weighted onℓ^p. Related results inner product and semi-inner product may also be found in [1, 7, 21, 22].

Here we shall formulate a new inner product usingn-norm on a real vector spaceX (dim(X)≥n). Recall that ann-normonXis a function∥·, . . . ,·∥:X×· · ·×X−→

Rwhich satisfies the following four properties:

(1) a1, . . . , an are linearly dependent if and only if∥a1, . . . , an∥= 0;

(2) ∥a1, . . . , an∥ invariant under permutation;

(3) ∥γa1, . . . ,n∥ = |γ|∥a1, . . . , an∥ for any any a1, . . . , an ∈ X and for every γ∈R;

(4) ∥a1, . . . , an−1, b+c∥ ≤ ∥a1, . . . , an−1, b∥+∥a1, . . . , an−1, c∥ for every b, c a1,...,an−1∈X.

Now we call that the pair (X,∥·, . . . ,·∥) is ann-normed space. Usually, the interpre- tation of ∥x₁,· · · , x_n∥ is the volume of the n-dimensional parallelepiped spanned a₁, ..., a_n ∈X. The development of the theory of 2-normed spaces was started since the late 1960’s. G¨ahler had an idea to generalize an area in a real vector space.

The theory ofn-normed spaces forn≥2 was developed in the late 1960’s [4, 5, 6].

Recent results can be found, for instance, in [2, 9, 13, 11, 17].

In this article, we will define an inner product on a subspace of (X,∥·,· · ·,·∥).

We also discuss the completeness of the subspace that equipped with the inner product. Motivated by this fact, we want to have a simple andgood definition of angle on a subspace ofn-normed space.

2. MAIN RESULTS

In this part, we can define an inner product on a subspace of (X,∥·,· · ·,·∥).

Next, we also discuss the completeness of the subspaces that equipped with this inner product.

2.1. A New Inner Product. Suppose that (X,∥·,· · ·,·∥) is a n-normed space.

Now, take a fixed set of linearly independent vectors

A={f1,· · ·, fn} ⊂X. (2)

Then we have that

Y := span(A) (3)

is a subspace of X. For g ∈ Y, there is ag = (a1g,· · · , ang) ∈ Rⁿ such that g=

n

P

i=1

aigfi.

On a normed space, there are many notions of orthogonality [3]. Two of them are Pythagorean orthogonality and isosceles orthogonality, as introduced by R.C.

James in [14]. Note that if the normed space is the inner product space, then the two types of orthogonality coincide with the definition of orthogonality on the inner product space. Moreover, we think that all the properties about orthogonality in a normed space can be applied to an n-normed space. Now, we give a quadratic formula onR.

Lemma 2.1. For every c1,· · ·, cn, d1,· · ·, dn∈R, we have

n

X

j=1

(c_j+d_j)²−





n

X

j=1

c²_i +

n

X

j=1

d²_i



= 2

n

X

j=1

c_id_i.

Proof. We givec_i, d_i∈Rforj= 1,· · ·, n. First, we observe that (c_j+d_j)²−c²_j−d²_j =c²_j +d²_j+ 2c_jd_j−c²_j−d²_j = 2c_jd_j, wherej = 1,· · · , n. Hence

n

P

j=1

(cj+dj)²−

n

P

j=1

c²_j+

n

P

j=1

d²_j

!

= 2

n

P

j=1

cjdj. □ In Lemma 2.1, if the real numbers above are viewed as vectorsx= (c1, . . . , cn), y= (d1, . . . , dn) inRⁿand

n

P

j=1

cjdj = 0, thenxandyare said to be orthogonal. As a result, the two vectors form a right triangle with hypotenusex+y. One may say thatxandyare orthogonal (Pythagorean type). The above lemma will be used to prove the following result.

Proposition 2.2. Let(X,∥·,· · ·,·∥)be ann-normed space, (2) and (3). Then we obtain

2





n

X

j=1

ajgajh



∥f1,· · ·, fn∥²= X

{j2,···,jn}⊆{1,···,n}

∥g+h, fj₂,· · ·, fj_n∥²

− X

{j2,...,jn}⊆{1,···,n}

∥g, fj₂,· · ·, fj_n∥²

− X

{j₂,···,j_n}⊆{1,···,n}

∥h, f_j2,· · ·, f_jn∥²

for every g, h∈Y.

Proof. Let (X,∥·,· · · ,·∥) be an n-normed space, (2) and (3). We observe that for α, β ∈ R, we have ∥αf1+βfj, f2,· · ·, fn∥ = |α|∥f1,· · ·, fn∥, for every j = 2,· · ·, n. Because Y := span{f1,· · · , fn} then for g ∈ Y, there is ag = (a1g,· · · , ang)∈Rⁿ such thatg=

n

P

i=1

ajgfi. Moreover,

∥g, f2,· · · , fn∥ =

n

X

i=1

aigfi, f2,· · ·, fn

= ∥a_1gf₁, f₂,· · · , f_n∥

= |a1g| ∥f1, f2,· · ·, fn∥. Consequently, for everyg, h∈Y we have

X

{j2,···,jn}⊆{1,···,n}

∥g, fj₂,· · ·, fj_n∥²=

n

X

j=1

a²_jg∥f1,· · · , fn∥², (4)

X

{j2,···,jn}⊆{1,···,n}

∥h, fj₂,· · ·, fj_n∥²=

n

X

j=1

a²_jh∥f1,· · ·, fn∥². (5) and

X

{j2,···,jn}⊆{1,···,n}

∥g+h, fj₂,· · ·, fj_n∥² =

n

X

j=1

(ajg+ajh)²∥f1,· · ·, fn∥². (6) Using (4), (5), (6) and by Lemma 2.1,

X

{j2,···,jn}⊆{1,···,n}

∥g+h, fj₂,· · ·, fj_n∥²− ∥g, fj₂,· · ·, fj_n∥²− ∥h, fj₂,· · ·, fj_n∥²

= 2





n

X

j=1

ajgajh



∥f1,· · ·, fn∥²

holds. The proof is complete. □

Another quadratic formula for real numbers is also presented in the following lemma.

Lemma 2.3. For every c₁,· · ·, c_n, d₁,· · ·, d_n∈R, we obtain

n

X

j=1

(c_j+d_j)²−(c_j−d_j)²

= 4

n

X

j=1

c_jd_j.

Proof. Suppose thatcj, dj∈Rforj= 1,· · · , n. We know that (c_j+d_j)²−(c_j−d_j)²= (c²_j+d²_j+ 2c_jd_j)−(c²_j+d²_j−2c_jd_j) = 2c_jd_j, j= 1,· · · , n. Consequently,

n

P

j=1

(c_j+d_j)²−(c_j−d_j)²

= 4

n

P

j=1

c_jd_j. □

Furthermore, like Lemma 2.1, the real numbers in the Lemma 2.3 can also be seen as two vectors x= (c1, . . . , cn) and y = (d1, . . . , dn) inRⁿ. For

n

P

j=1

cjdj = 0, we have

n

X

j=1

(c_j+d_j)²=

n

X

j=1

(c_j−d_j)²

or we write thatxandy are orthogonal isoceles. Next, in an n-normed space, we use Lemma 2.3 to explain the following proposition.

Proposition 2.4. If (X,∥·,· · · ,·∥)be an n-normed space, (2) and (3), then 4





n

X

j=1

ajgaih



∥f1,· · ·, fn∥²= X

{j2,···,jn}⊆{1,···,n}

∥g+h, fj₂,· · · , fj_n∥²

− X

{j2,···,j_n}⊆{1,···,n}

∥g−h, fj₂,· · · , fj_n∥²

for any g, h∈Y.

Proof. By the above assumptions and the property

∥αf1+βfj, f2,· · ·, fn∥=|α|∥f1,· · · , fn∥, withα, β∈Rand for anyj= 1,· · ·, n, we obtain that

∥g, f2,· · ·, fn∥=|a1g| ∥f1, f2,· · · , fn∥ holds for every g∈Y. Now check that

X

{j2,···,jn}⊆{1,···,n}

∥g+h, f_j2,· · ·, f_jn∥² =

n

X

j=1

(a_jg+a_jh)²∥f₁,· · ·, f_n∥², (7) for every g, h∈Y, and

X

{j2,···,jn}⊆{1,···,n}

∥g−h, fj₂,· · ·, fj_n∥² =

n

X

j=1

(ajg−ajh)²∥f1,· · ·, fn∥². (8) for every g, h∈Y. Next, we use (7), (8), and Lemma 2.3 to obtain

X

{j2,···,j_n}⊆{1,···,n}

∥g+h, fj₂,· · · , fj_n∥²− ∥g−h, fj₂,· · · , fj_n∥²

= 4





n

X

j=1

ajgajh



∥f1,· · ·, fn∥²

for anyg, h∈Y. □

Now, recall (2) and (3) to define

⟨g, h⟩_A:=1 4

X

{j2,···,jn}⊆{1,···,n}

∥g+h, fj2,· · · , fjn∥²− ∥g−h, fj2, . . . , fjn∥²

=





n

X

j=1

ajgajh



∥f1,· · ·, fn∥². (9)

for every g, h ∈ Y. The above functional is defined using Proposition 2.4. It is also similar topolarization law, while we want to check a norm (derived by an inner product or not). Another way, one may use Proposition 2.2 to obtain (9).

Now, note that the coefficients of the linear combination of the vectors inY can be viewed as an element ofRⁿ. For example, the coefficients ofg∈Y can be seen as ag = (a1g,· · · , ang)∈ Rⁿ. Thus, the relation between (1) and (9) is expressed as follows

⟨g, h⟩_A=⟨ag, ah⟩

Rⁿ∥f1,· · ·, fn∥², (10) for every g, h ∈ Y. Furthermore using (1), we shall prove that (10) is an inner product as follows.

Theorem 2.5. If(X,∥·,· · ·,·∥)is an-normed space, (2) and (3), then the mapping

⟨·,·⟩_A is an inner product onY.

Proof.Using all of the assumptions of this proposition,⟨·,·⟩_Awill be checked that it satisfies the properties of inner product.

(1) (non negative) Takeg∈Y. We have⟨a_g, a_g⟩

Rⁿ≥0 and∥f₁,· · ·, f_n∥>0, so we obtain⟨g, g⟩_A=⟨ag, ag⟩

Rⁿ∥f1,· · · , fn∥²≥0.

(⇒) We give⟨g, g⟩_A=⟨ag, a_g⟩

Rⁿ∥f1,· · ·, f_n∥²= 0. Since∥f1,· · · , f_n∥>

0, then

⟨ag, ag⟩

Rⁿ =

n

X

i=1

a²_ig

!

= 0.

Consequently, we obtainajg = 0 for everyj= 1,· · ·, n. So,g=

n

P

j=1

ajgfj = 0 holds.

(⇐) Suppose thatg= 0. Sinceag= 0∈Rⁿ, then ⟨ag, ag⟩

Rⁿ = 0 holds.

We obtain⟨g, g⟩_A= 0.

(2) (commutative) Takeg, h∈Y. Check that

⟨g, h⟩_A=⟨a_g, a_h⟩

Rⁿ∥f₁,· · ·, f_n∥²=⟨a_h, a_g⟩

Rⁿ∥f₁,· · ·, f_n∥²=⟨h, g⟩_A. (3) (homogen) Takeg, h∈Y andα∈R. We have

⟨αg, h⟩_A=⟨αag, a_h⟩

Rⁿ∥f1,· · · , f_n∥²

=α⟨ag, ah⟩

Rⁿ∥f1,· · · , fn∥²

=α⟨g, h⟩_A.

(4) (distributive) Takeg₁, g₂, h∈Y. We observe that

⟨g1+g2, h⟩_A=⟨ag1+ag2, ah⟩

Rⁿ∥f1, . . . , fn∥²

=⟨ag₁, ah⟩

Rⁿ∥f1,· · · , fn∥²+⟨ag₂, ah⟩

Rⁿ∥f1,· · ·, fn∥²

=⟨g₁, h⟩_A+⟨g₂, h⟩_A.

Hence,⟨·,·⟩_A is an inner product onY. □

We also give a norm that is inducted by the inner product onY. Corollary 2.6. Letsg∈Y. The following

∥g∥_A:=

q

⟨ag, ag⟩

Rⁿ∥f1,· · ·, fn∥=∥ag∥_Rⁿ∥f1,· · · , fn∥ (11) defines the norm on Y.

Because Y = span (A) with A = {f₁,· · ·, f_n} then we have the following corollary.

Corollary 2.7. In(Y,⟨·,·⟩_A), setA={f1,· · · , f_n}is an orthogonal set respect to

⟨·,·⟩_A.

2.2. The Completeness of subspaces of n-Normed Space. We have that (Y,⟨·,·⟩_A) is an inner product space. Using (10), all of the properties of (Rⁿ,⟨·,·⟩

Rⁿ) can be delivered to (Y,⟨·,·⟩_A). Note that each norm definition inRⁿ is equivalent.

Meanwhile, Rⁿ as a normed space is a complete space. The completness of Y of n-normed SpaceX as follows.

Theorem 2.8. The subspace (Y,∥·∥_A)is a complete space.

Proof. Let (w_k) be a Cauchy sequence inY. Hence, wk =a1w_kf1+· · ·+anw_kfn.

It means that for any ϵ >0, there is an n^′_ϵ ∈N such that for everyk, l > n^′_ϵ, we have

∥w_k−w_l∥_A=∥a_wk−a_wl∥

Rⁿ∥f₁,· · · , f_n∥< ϵ.

As consequence, we obtain

∥aw_k−aw_l∥_Rn< ϵ

∥f1,· · ·, fn∥ =ϵ^′.

We say thata_wk ∈Rⁿ is also a Cauchy sequence. BecauseRⁿ is a complete space respect to ∥·∥

Rⁿ, thena_wk ∈Rⁿ is a convergence sequence. Clearly, (w_k) in Y is also a convergence sequence. Hence, (Y,∥·∥_A) is a complete space. □ By Theorem 2.8, we also conclude that (Y,∥·∥_A) is a Banach space. Moreover, here∥·∥_Ais induced by⟨·,·⟩_A, so (Y,⟨·,·⟩_A) is a Hilbert space. On (Y,⟨·,·⟩_A), several

functionals can be defined (see some books of functional analysis). In particular, to define them-inner product, we have to usem < n.

Here, we note that Y = span (A) and an orthogonal set A = {f₁, . . . , f_n} on (X,∥·, . . . ,·∥) respect to ⟨·,·⟩_A (see Corollary 2.7). Now, we give a linearly independent setB={g1, . . . , gn}onY. One may form

g1=

n

X

i=1

ai g₁fi

g2=

n

X

i=1

ai g₂fi

... gn=

n

X

i=1

ai g_nfi

or B =KA, where B =









 g₁ g₂ ... gn









 , K =











a_1g1 a_2g1 · · · a_{n g1} a_1g2 a_2g2 · · · a_{n g2} ... ... . .. ... a1g_n a2g_n · · · an g_n











and A=









 f₁ f₂ ... fn









 .

It is obvious that K is an invertible matrix, soK⁻¹B =A holds. It means that set A can also be developed by set B. With an initial ¯g1 = _∥g^g1

1∥_A, we obtain an orthonormal set ¯B ={g¯₁,g¯₂,· · · ,g¯_n} by using the Gram-Schmidt process respect to⟨·,·⟩_A. Consequently,

Y = span (A) = span (B) = span ( ¯B).

3. FURTHER RESULTS

Here, we recall (Y,⟨·,·⟩_A). For any p, q ∈ Y, we write p =

n

P

j=1

aj pfj and q=

n

P

j=1

a_{j q}f_j wherea_p= (a_1p,· · ·, a_{n p}), a_q= (a_1q,· · ·, a_{n q})∈Rⁿ. Next, we have the angle between pandq(ϕ(p, q)) of (Y,⟨·,·⟩_A) as follows.

cos²ϕ(p, q) = ⟨p, q⟩²_A

∥p∥²_A∥q∥²_A = ⟨ap, aq⟩²

Rⁿ∥f1,· · · , fn∥⁴

∥ap∥²

Rⁿ∥aq∥²

Rⁿ∥f1,· · · , f_n∥⁴ = ⟨ap, aq⟩²

Rⁿ

∥ap∥²

Rⁿ∥aq∥²

Rⁿ

. (12) Hence, the angleϕbetween two vectorspand qin (Y,⟨·,·⟩_A) is equivalen to the angle between two vectors a_panda_q in (Rⁿ,⟨·,·⟩).

Moreover, we can formulate angleϕbetween two subspaces onY. The result is shown as follows.

See in [12] and now letP = span{p₁,· · ·, p_n1} andQ= span{q₁,· · · , q_n2}be subspaces ofY, with 1≤n1≤n2 and {p1,· · ·, pn₁}, {q1,· · ·, qn₂}are orthogonal sets. Now, we have the angle ϕbetweenP danQ

cos²ϕ(P, Q) = det(M^TM) (13)

where

M^T =













⟨p1,q1⟩_A

∥p₁∥_A∥q₁∥_A

⟨p1,q2⟩_A

∥p₁∥_A∥q₂∥_A · · · ⟨^p1,qn2⟩_A

∥p1∥_A∥^qn2∥_A

... ... . .. ...

⟨^pn1,q1⟩_A

∥^pn1∥_A∥q1∥_A

⟨^pn1,q2⟩_A

∥^pn1∥_A∥q2∥_A · · · ⟨^pn1,qn2⟩_A

∥^pn1∥_A∥^qn2∥_A











 .

Since _∥p^⟨pj,qk⟩A

j∥_A∥qk∥_A = ⟨^a_pj^,a_qk⟩

Rn∥f₁,···,f_n∥²

∥^a_pj∥

Rn∥^a_qk∥

Rn∥f₁,···,f_n∥² = ⟨^a_pj^,a_qk⟩

Rn

∥^a_pj∥

Rn∥^a_qk∥

Rn

where j : 1,· · ·, n1

andk: 1,· · ·, n2, then

M^T =













⟨^ap1,aq1⟩

Rn

∥^ap1∥

Rn∥^aq1∥

Rn

⟨^ap1,aq2⟩

Rn

∥^ap1∥

Rn∥^aq2∥

Rn

· · · ⟨^ap1,a_qn

2⟩

Rn

∥^ap1∥

Rn∥^aqn2∥

Rn

... ... . .. ...

⟨^apn1,a_q1⟩

Rn

∥^apn1∥

Rn∥^aq1∥

Rn

⟨^apn1,a_q2⟩

Rn

∥^apn1∥

Rn∥^aq2∥

Rn

· · · ⟨^apn1,a_qn

2⟩

Rn

∥^apn1∥

Rn∥^aqn2∥

Rn











 .

We know that (12) is a spesial case of (13).

Acknowledgement. The research is supported by LPPM Unhas Collaborative Fundamental Research Program 2023 No. 00323/UN4.22/PT.01.03/2023.

REFERENCES

[1] V. Balestro, `A.G. Horv`ath, H. Martini, and R. Teixeira, Angles in normed spaces,Aequationes Math.,91(2)(2017), 201-236.

[2] H. Batkunde, and H. Gunawan, A revisit ton-normed spaces through its quotient spaces, Matematychni Studii53(2)(2020), 181-191.

[3] R. Benitez, Carlos Orthogonality in normed linear spaces: a classification of the different concepts and some open problems,Revista Mathematica2(1989) 53-57.

[4] S. G¨ahler, Lineare 2-normierte r¨aume,Math. Nachr.28(1964), 1-43.

[5] S. G¨ahler, Untersuchungen ¨uber verallgemeinertem-metrische R¨aume. I,Math. Nachr.40 (1969), 165-189.

[6] S. G¨ahler, Untersuchungen ¨uber verallgemeinertem-metrische R¨aume. II,Math. Nachr.40 (1969), 229-264.

[7] J. R. Giles, Classes of semi-inner-product spaces,Trans. Amer. Math. Soc.129(3)(1967), 436-446.

[8] H. Gunawan, On n-inner products, n-norms, and the Cauchy-Schwarz inequality,Sci. Math.

Jpn.55(2002), 53-60.

[9] H. Gunawan and M. Mashadi, Onn-normed spaces, Int. J. Math. Math. Sci.,27(2001), 321-329.

[10] H. Gunawan, O. Neswan, and W. Setya-Budhi, A formula for angles between two subspaces of inner product spaces,Beitr. Algebra Geom.,46(2)(2005), 311-320.

[11] H. Gunawan, O. Neswan, and E. Sukaesih, Fixed point theorems on bounded sets in an n-normed space,J. Math. Comput. Sci.,8(2)(2018), 196-215.

[12] H. Gunawan and O. Neswan, On angles between subspaces of inner product spaces, J. Indo.

Math. Soc.,11(2005), 129-135.

[13] M. Idris , S. Ekariani and H. Gunawan, On the space ofp-summable sequences, Mat. Vesnik., 65(1)(2013), 58-63.

[14] R.C. James, Orthogonality in normed linear spaces, Duke Math. J.12, (1945) 291-302.

[15] S. Konca, M. Idris, and H. Gunawan,p-summable sequence spaces with inner products,Beu J. Sci. Techn.,5(1)(2015), 37-41.

[16] E. Kreyszig,Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., New York, 1978.

[17] A.L. Soenjaya, Onn-bounded andn-continuous operator inn-normed space, Journal of the Indonesian Mathematical Society,18(1)(2012), 45-56.

[18] P. M. Mili`ci`c, On theB-angle andg-angle in normed spaces,J. Inequal. Pure Appl. Math., 8(3)(2007), 1-9.

[19] A. Misiak,n-inner product spaces,Math. Nachr.140(1989), 299-319.

[20] M. Nur, M. Idris and Firman, Angle in the space ofp-summable sequences,AIMS Mathe- matics,7(2)(2022), 2810-2819.

[21] M. Nur, and H. Gunawan, A new orthogonality and angle in a normed space,Aequationes Math.93(2019), 547-555.

[22] M. Nur, and H. Gunawan, A note on the g-angle between subspaces of a normed space, Aequationes Math.,95(2021), 309-318.