Many of the philosophical questions arose in the 17th century, during the early development of classical mechanics. In Isaac Newton's view, space was absolute - in the sense that it existed permanently and independently of whether there were any matter in the space.[2] Other natural philosophers, notably Gottfried Leibniz, thought instead that space was a collection of relations between objects, given by their distance and direction from one another. In the 18th century, Immanuel Kant described space and time as elements of a systematic framework which humans use to structure their experience.
In the 19th and 20th centuries mathematicians began to examine non-Euclidean geometries, in which space can be said to be curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space.[3] Experimental tests of general relativity have confirmed that non-Euclidean space provides a better model for explaining the existing laws of mechanics and optics.
Hilbert spaces:
Link:
http://en.wikipedia.org/wiki/Hilbert_spaceDescription:
The
mathematical concept of a
Hilbert space, named after
David Hilbert, generalizes the notion of
Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. In more formal terms, a Hilbert space is an
inner product space — an abstract
vector space in which distances and angles can be measured — which is "
complete", meaning that if a sequence of vectors is
Cauchy, then it converges to some limit in the space.
Euclidean space:
Link:
http://en.wikipedia.org/wiki/Euclidean_spaceDescription:Around 300
BC, the
Greek mathematician Euclid undertook a study of relationships among
distances and
angles, first in a plane (an idealized flat surface) and then in space. An example of such a relationship is that the sum of the angles in a
triangle is always 180
degrees. Today these relationships are known as two- and three-
dimensional Euclidean geometry. In modern
mathematical language, distance and angle can be generalized easily to 4-dimensional, 5-dimensional, and even higher-dimensional spaces. An
n-dimensional space with notions of distance and angle that obey the Euclidean relationships is called an
n-dimensional
Euclidean space.
Eigenvalue, Eigenvector, Eigenspace:
Link:
http://en.wikipedia.org/wiki/EigenvalueDescription:
In
mathematics, given a
linear transformation, an
eigenvector (help·info) of that linear transformation is a nonzero
vector which, when that transformation is applied to it, may change in length, but it remains along the same line [the direction will "flip" if the eigenvalue is negative].
For each eigenvector of a linear transformation, there is a corresponding scalar value called an
eigenvalue (info) for that vector, which determines the amount the eigenvector is scaled under the linear transformation. For example, an eigenvalue of +2 means that the eigenvector is doubled in length and points in the same direction. An eigenvalue of +1 means that the eigenvector is unchanged, while an eigenvalue of −1 means that the eigenvector is reversed in sense. An
eigenspace of a given transformation for a particular eigenvalue is the set (
linear span) of the eigenvectors associated to this eigenvalue, together with the
zero vector (which has no direction).