The concept of linear subspace or vector subspace as it is sometimes called is very important in linear algebra and various other fields of physics and math. A linear subspace is commonly referred to as subspace when the context serves to distinguish it from other kind of subspaces. The subspace proof explained.

Let K be the field and V be the vector space over K. Let the element of V be vectors and that of K be scalars. Now let W be the subset of V. If W is itself the vector space with the same vector space operations as V has then it is known as a subspace of V.

In order to use this definition and as a subspace proof we need not prove that all the propertites of vector space hold for W. Instead of it, we can prove a theorem that will give us an easier way to show that a subset of vector space is a subspace.

Assume that V is the vector space over the field K and also assume that W is the subset of V. Then we can say that W is a subset when

- The zero vector is in W.
- Let U and V are the elements of W, then the sum U+V is an element of W.
- If C is a scalar from K and U is an element of W, then the scalar product, i.e. CU is an element of W.

Proof
The first property confirms that W is non- empty. Also the second and third property assumes that closure of W under the addition and the scalar multiplication; therefore the vector space orientations are well defined.

The elements of W are also elements of V, so axioms 1-2 and 5-8 of the vector space are satisfied. By closing W under scalar multiplication the axioms 3 and 4 of a vector space are satisfied.

Conversely if W is a subspace of V. Then W itself will become a vector space under the various operations induced by V. As a result of it the second and third properties are also satisfied. By the virtue of property third, -w is in W wherever w is. And it also follows that W is closed under subtraction as well. Now since W is non empty therefore there is an element x in W, and x-x=0 is also in W. Therefore property one is also satisfied.

In the Star Trek fictional universe, sub space is a feature of space- time which facilitates faster than light transit in the form of transmission of information or interstellar travel. The Star Trek Subspace follows different laws of physics. Subspace has also been adopted and used in various series and video games like The Descent, Stargate Atlantis, Halo and Freespace.

In majority of the Star Trek series subspace is mainly used for establishing contacts with the people and places which are light years away. The physics of the Star Trek Subspace is described as infinite speed which is expressed as Warp 10, as an impossible thing, as such even the subspace communications which supposedly travel at speeds over 9.9 Warp may take weeks to reach a desired destination and are confined to region of space which was mapped previously for subspace channels. However, once the connection is established then the communications between the two points become instantaneous.

Subspace is basically a substrate within which our universe exists. A subspace filed may either be a natural or forced intrusion of this domain into our space, thus altering the behaviors of things which are in our time frame.

There are many things which support this. In “Schisms” different creatures exist in within a tertiary subspace manifold. A manifold is a term which is used to describe the form our universe takes when it is viewed from a theoretical direction, that is, a higher direction. This is also sometimes called as the “deeper level of subspace”.

Whenever our space time frame is distorted a large amount of energy is released. You can call subspace as a medium in which our space- time frame exists. The parts which are nearest are coupled tightly to our own universe. And can be thought of as being mapped to our own space-time.

Subspace is not a ‘place’ or an alternate reality or space-time frame where things go. Subspace is also not entered by a starship at warp. A ship only creates a subspace which acts like another universe very tightly coupled to our own.

Subspaces do not really need to follow the rules of relativity. These might have unique reference frame, and everything which is enclosed in a subspace field has the reference frame of the subspace.

The weakest sub space fields are just like traditional fields, like magnetic fields. They consist of associated particles, can be used for transmission, can be bound to objects etc.