2.1. Semi-Inner Product and -Angle between Two Vectors
In this subsection, we shall begin with the new semi-inner product on
spaces equipped with an
n-norm. Let
be an
n-normed space and
be a linearly independent set on
. Now, we define the following mapping.
for every
. Next, we have the following proposition.
Proposition 1 ([
17])
. The mapping defines a norm on . Example 1. For . Let be the Schauder basis on , that is, and . We observe that Using the norm
with
for
, we define a mapping
on the
n-normed space
with
by
for every
Then we have the following result.
Theorem 1. The mapping in (4) defines a semi-inner product on . Proof. We will verify that satisfies the properties (1–4) of the semi-inner product.
Using the properties of the determinant, we have
Therefore, defines a semi-inner product on . □
Example 2. For . Let be the Schauder basis on , that is, and in Proposition 1. If and in , we haveand Remark 1. The function does not satisfy commutative property. In the example above, we observe that .
Specifically for
, we observe that
As consequence, we have following corollary:
Corollary 1. The mapping in (4) defines an inner product on . Example 3. For . Let be the Schauder basis on , that is, . If and in , we have By using the semi-inner product , we define orthogonality in as follows:
Definition 1 (-orthogonality). Let be the n-normed space. Vector x is -orthogonal to vector y, and we write if and only if .
Example 4. For . Let be the Schauder basis on , that is, . If and in , using Example 2, we have . Hence, x is -orthogonal to y.
Next, -orthogonality has the following properties.
Proposition 2. The -orthogonality satisfies the following properties:
- (a)
Nondegeneracy property: If , then .
- (b)
Homogeneity property: If , then for every .
- (c)
Right additive property: If and , then .
- (d)
Resolvability property: For every there is such that .
Proof. By using Theorem 1, the properties (a)−(c) are obviously true.
- (d)
Let
. For case
, the resolvability property is fulfilled. Next, for case
, choose
. Using Theorem 1, we have
as desired.
□
Remark 2. Note that for , the -orthogonality coincides with the g-orthogonality in . Specifically for , the -orthogonality satisfies the symmetry and continuity property. Next, by using Remark 1, the -orthogonality does not satisfy the symmetry property. The -orthogonality also does not satisfy continuity property.
Example 5. For . Take , , and in . Using inequality in [17], we have (in norm ). Next, we observe that for every , but . Using a semi-inner product in (
4), we define the angle between two nonzero vectors
x and
y on
as follows:
Note that
if and only if
We can observe that the angle
for
is identical with the
g-angle
in [
9].
Example 6. Let be 3 normed space and be the Schauder basis on , that is, and in Proposition 1. If and in , we observe thatandAs a consequence,Hence, . The angle has the following properties.
Proposition 3. Let be the n-normed space. The angle between two nonzero vectors x and y on satisfies the following properties:
- (a)
If x and y are of the same direction, then ; if x and y are of the opposite direction, then (part of parallelism property).
- (b)
if ; if (homogeneity property).
- (c)
If (in norm ), then (part of continuity property).
Proof. - (a)
Let
for an arbitrary nonzero vector
x in
X and
We have
Hence,
for
and
for
- (b)
Let
and
Observe that
If
, then
Likewise, if
, then
.
- (c)
If
(in norm
), then
Observe that
. We have
. Hence,
as desired.
□
Remark 3. Since the mapping in general is not commutative, the angle does not satisfy the symmetry property. For example, we can see Remark 1. Likewise, the g-angle does not satisfy the continuity property.
Example 7. For and . Take , , and in . We observe that for every , but .
2.2. -Angle between Two Subspaces
In this section, we will discuss the
-angle between two subspaces in
. In particular, we define the
-angle between a one-dimensional subspace and an
m-dimensional subspace for
. Using
in (
4), we have the Gram determinant
as follows.
Definition 2. Let be an n-normed space and fo is subspace of . The Gram determinant of , denoted by is defined by Example 8. For and . Let be the Schauder basis on , that is, and in . We observe that and . Using the semi-inner product in (4), we haveandHence, . We have the connection between the Gram determinant of for and the linearly independence set of as follows.
Theorem 2. If then is a linearly independent set.
Proof. Suppose by contradiction that is linearly dependent. Therefore, there is a i with so that is a linear combination of . Using the properties of the determinant, we observe that the i-th column of is a linear combination of the other columns. This implies , which is a contradiction. Hence, is a linearly independent set. □
Example 9. Suppose that , . Let be the Schauder basis on , that is, and in Proposition 1. If and in . We observe that and As a consequence,In a similar way is obtained . As a consequence, Hence, if then is a linearly independent set. Remark 4. The converse of the above theorem is not true. For instance, let be a 2-normed space with and . Take and in . We observe that and . Using the semi-inner product in (4), we haveandHence, but and are linearly independent. Next, we can discuss the -orthogonal projection and the - orthogonal complement as follows:
Definition 3. Let v be a vector of and be a subspace of with . The -orthogonal projection of v on W, denoted by , is defined byand its -orthogonal complement is given by Example 10. Suppose that , and . Let be the Schauder basis on , that is, and in Proposition 1. If and , then by Example 9. Therefore,and Using the properties of the determinant and the semi-inner product in (
4), we obtain
Hence,
for every
.
The definition of the -angle between and of is as follows.
Definition 4. Let be an n-normed space for . If and are subspaces of with dan then the -angle between subspaces V and W is denoted by withwhere denotes the -orthogonal projection of v on W. Example 11. Suppose that , and in Definition 4 Let be the Schauder basis on , that is, and in Proposition 1. If and with and in , we observe thatBy using Example 9, we have Therefore,Moreover, we observe that and . Hence, Remark 5. If , then the -orthogonal projection of v on W isUsing this -orthogonal projection, the properties of the semi-inner product and Definition 4, we have the -angle between subspaces V and W, that is,Hence, we can see that the -angle between the two vectors is also the -angle between these two vectors in the subspace spanned by them. Next, by using Definition 3, we have . Therefore, Definition 4 may be rewritten asThis tells us that the value of is equal to the ratio between the norm of the -orthogonal projection of v on W and the norm of v.