Abstract
As discussed in the previous chapter, vector spaces are analogous to three-dimensional Euclidean space \(\mathbf {V}_3\) of geometric vectors (directed line segments). However, such important concepts as the length of a vector and the angle between two vectors were not introduced for abstract spaces. In three-dimensional Euclidean space, using the lengths of two vectors and the angle between them, we can calculate the inner product (the dot product) of these vectors. Many geometric problems in the spaceĀ \(\mathbf {V}_3\) are solved with help of the dot product. The concept of an inner product on an abstract space will be introduced axiomatically in this chapter. After that, the concepts of the length of a vector and the angle between two vectors will be introduced based on the concept of the inner product. Then we will investigate the concept of orthogonal bases. Some important examples of orthogonal bases in finite-dimensional spaces, particularly in polynomial spaces, will be constructed. The basic properties of subspaces of unitary spaces will be described. We begin our considerations with inner products on the spaces \({\mathbb {R}}^n\) andĀ \({\mathbb {C}}^n\).
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Notes
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The verification of inequality 3 will be carried out in SectionĀ 3.2.2, p.Ā 72.
- 2.
Hermann Minkowski (1864ā1909) was a German mathematician.
- 3.
We can say that the vector b is the projection of the vector a on the line that is parallel to the vectorĀ b.
- 4.
Pythagoras of Samos (570ā495 B.C.) was an Ionian Greek philosopher and mathematician.
- 5.
Augustin-Louis Cauchy (1789ā1857) was a French mathematician, Karl Hermann Amandus SchwarzĀ (1843ā1921) was a German mathematician.
- 6.
JĆørgen Pedersen Gram (1850ā1916) was a Danish mathematician.
- 7.
Erhard Schmidt (1876ā1959) was a German mathematician.
- 8.
Adrien-Marie Legendre (1752ā1833) was a French mathematician.
- 9.
Benjamin Olinde Rodrigues (1794ā1851) was a French mathematician.
- 10.
Jean Baptiste Joseph Fourier (1768ā1830) was a French mathematician and physicist.
- 11.
Pafnuty Lvovich Chebyshev (1821ā1894) was a Russian mathematician.
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Friedrich Wilhelm Bessel (1784ā1846) was a German mathematician and astronomer.
- 13.
David Hilbert (1862ā1943) was a German mathematician.
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Beilina, L., Karchevskii, E., Karchevskii, M. (2017). Inner Product Spaces. In: Numerical Linear Algebra: Theory and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-57304-5_3
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DOI: https://doi.org/10.1007/978-3-319-57304-5_3
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