A formula for the g-angle between two subspaces of a normed space

Abstract

We develop the notion of g-angle between two subspaces of a normed space. In particular, we discuss the g-angle between a 1-dimensional subspace and a t-dimensional subspace for t ≥ 1 and the g-angle between a 2-dimensional subspace and a t-dimensional subspace for t ≥ 2. Moreover, we present an explicit formula for the g-angle between two subspaces of ℓ p spaces.

A New Semi-Inner Product and pn-Angle in the Space of p-Summable Sequences

“…Note that y ⊥ p n x if and only if A p n (x, y) = 1 2 π. We can observe that the angle A p n (x, y) for n = 1 is identical with the g-angle A g (x, y) in [9] . Example 6.…”

“…Many researchers have studied the g-orthogonal and g-angle between two vectors and two subspaces in X; see, for example, [5][6][7][8]. In 2018, Nur et al [9] developed the notion of the g-angle between two subspaces. If V = span{v} and W = span{w 1 , • • • , w m } of X with m ≥ 1, then the g-angle between V and W is defined by A g (V, W) with…”

Three Equivalent n-Norms on the Space of p-Summable Sequences

“…One may then verify that the three 2-norms •, • 2 , •, • 2 and •, • g,2 are identical (see [6,12]).…”

“…By using a semi-inner product g, Miličić [3] introduced the following orthogonality relation on X: x is said to be g-orthogonal to y, denoted by x ⊥ g y, provided that g(x, y) = 0. For more recent works, see in [4,5]. Recently, Nur and Gunawan in [6] defined a 2-norm on X by x 1 , x 2 g := sup y j ≤1, j=1,2 g(y 1 , x 1 ) g(y 2 , x 1 ) g(y 1 , x 2 ) g(y 2 , x 2 ) .…”

“…For more recent works, see in [4,5]. Recently, Nur and Gunawan in [6] defined a 2-norm on X by x 1 , x 2 g := sup y j ≤1, j=1,2 g(y 1 , x 1 ) g(y 2 , x 1 ) g(y 1 , x 2 ) g(y 2 , x 2 ) .…”