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: WikiVector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spacesand Hilbert spaces, which are fundamental in mathematical analysis.

: WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.

I can't answer your OP question Jeff, but I am curious as to this concept of vector spaces and can at least stay 'near' topic...Quote : WikiVector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spacesand Hilbert spaces, which are fundamental in mathematical analysis....andQuote : WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.Could g (gravity) and a (acceleration) be described as force vectors of the same type?Or is a (acceleration) a 'real multiplier'?

: WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector

It's in the Wiki quote:Quote: WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector

http://www.einstein-online.info/spotlights/gravity_of_gravityThis gives clear indication as to why you might be asking such a question...

Is there a vector space that can be used with linear combinations that is representative of a non-linear space such as that of the gravitational field? If this exists can it be formulated as an energy vector space?

Quote from: jeffreyH on 16/01/2016 18:26:53Is there a vector space that can be used with linear combinations that is representative of a non-linear space such as that of the gravitational field? If this exists can it be formulated as an energy vector space?Jeffrey, gravity fields and curvature of space is only curved relative to ''flat'' space/background being linear.

Quote from: Thebox on 01/04/2017 21:17:53Quote from: jeffreyH on 16/01/2016 18:26:53Is there a vector space that can be used with linear combinations that is representative of a non-linear space such as that of the gravitational field? If this exists can it be formulated as an energy vector space?Jeffrey, gravity fields and curvature of space is only curved relative to ''flat'' space/background being linear. Yes. That is very perceptive.Quote from: jeffreyH on 01/04/2017 21:21:30Quote from: Thebox on 01/04/2017 21:17:53Quote from: jeffreyH on 16/01/2016 18:26:53Is there a vector space that can be used with linear combinations that is representative of a non-linear space such as that of the gravitational field? If this exists can it be formulated as an energy vector space?Jeffrey, gravity fields and curvature of space is only curved relative to ''flat'' space/background being linear. Yes. That is very perceptive.Any fields or waves from (a) to (b) exist in the present.

Quote from: Thebox on 01/04/2017 21:17:53Quote from: jeffreyH on 16/01/2016 18:26:53Is there a vector space that can be used with linear combinations that is representative of a non-linear space such as that of the gravitational field? If this exists can it be formulated as an energy vector space?Jeffrey, gravity fields and curvature of space is only curved relative to ''flat'' space/background being linear. Yes. That is very perceptive.