Subspaces of a Topological Vector Spaces I have a few questions about topological spaces which I am currently studying.
First some definitions that I am using:
Definition of subspace topology:
Given a topological space $(X,\tau)$ and a subset $S$ of $X$, the subspace topology on $S$ is defined by $\tau_{S} = \{S \cap U : U \in \tau \}$. 
Definition of linear subspace:
I use the usual definition, contains the zero vector and closed under addition and scalar multiplication.
Definition of topological vector space:
A vector space $X$ over a field $K$ which is endowed with a topology such that vector addition $X \times X \rightarrow X$ and scalar multiplication $K \times X \rightarrow X$ are continuous functions(where $X \times X$ endowed with product topology).
Questions:


*

*Why is the topological vector space defined in this way?

*Is a linear subspace in a topological vector space(e.g. normed space) automatically a subspace of the topological vector space or does it require additional properties?

*Does a subspace of a topological vector space satisfy the definition of subspace as stated in my first definition? How would I show this?

*Consider a normed vector space $X$ and a subset $W$. If we endow $W$ with a different norm that that of $X$ then what are the requirements necessary for $W$ to be a subspace of $X$?


Thanks a lot for any assistance!  
 A: You're using the word 'subspace' to mean two different things. When we talk about the 'subspace topology', we just mean endowing any subset of the topological space $X$ with a topology making it a topological space (hence, subspace). However, the definition of linear subspace is "subspace closed under the operations of linear algebra" - that is, a subspace of a vector space. Now, a topological vector space is two things at once - but above all, we always want it to be a vector space. So when we say the subspace of a topological vector space, we mean it in both ways at once - it's a subspace (in the sense of linear algebra) endowed with the subspace topology (making it into a topological space) - so a subspace of a topological vector space is also a topological vector space.
Re: #4: When we talk about a subspace of a topological vector space, we are specifically endowing a linear subspace with the subspace topology - so if the new norm induces a different topology, it's just another topological vector space, no mention made of our original one.
A: As the previous responder already highlighted, there are several things at play here. Take a linear subspace of a topological vector space. By this we mean a set closed w.r.t. finite linear combinations. Then endow it with the subspace topology (this is an example of initial topology - it automatically makes it into a topological space https://en.wikipedia.org/wiki/Initial_topology). Now you have two structures on your subset:


*

*A linear subspace

*A topological space


What makes a topological vector space special? The compatibility of these two structures, i.e.:


*

*continuity of the addition operation

*continuity of the scalar multiplication


But since the restriction of a continuous map is a continuous map (https://proofwiki.org/wiki/Restriction_of_Continuous_Mapping_is_Continuous). The continuity of the vector addition and scalar multiplication gives the continuity on of these operations on your linear subspace. Thus making it into a topological vector space. Notice how the linearity of the subspace plays an important role as we never "drop out" of the space when performing addition or multiplication. Hope this helps!
