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!
Best Answer
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.