Some Questions on Topological vector spaces

Question

I am having really hard time in my Functional Analysis II class. I never had this problem with Functional Analysis I class, which mostly focused on metrics spaces, normed spaces, banach space and hilbert space. In this class, we are mostly focused on topological vector space. I just find this space so confusing and hard to understand. I have some questions which I think will help me understand what is going on

Suppose that $X$ is a topological vector space (T.V.S) over field $F$

1. Does this mean that If $x,y\in X$, then $x+y\in X$ (linear space) and If $A,B\subseteq X$, then $A\cup B\subseteq X$ (topology)?
2. Why is it mentioned in some textbooks that the topology over $F$ is euclidean topology? is this the only topology we can have?
3. Does it mean that $F$ can only have euclidean topology while $X$ can have it is own separate and different topology?
4. Does T.V.S have to be Hausdorff? If no, then why is Hausdorff used in most theorems in our lecture notes? What is different about a non-Hausdorff T.V.S?
5. Does T.V.S imply that addition and multiplication operators are continuous or is it a necessary condition?
6. Why is a linear space with a discrete topology is not a topological linear space?

Sorry if these questions look trivial, but they are just not clear to me.

Answer 1

Here are some answers with references to this forum:

1.- Always is true that if $X$ is a topological vector space and $x,y\in X$ then $x+y\in X$ because $X$ is a vector space and the sum of vectors in a vector space is closed. The another condition ($A,B\subseteq X$ then $A\cup B\subseteq X$) is confusing because that property holds always by the definition of union. Maybe you refering to the topological structure?

2.- Not in general but the euclidean topology over the field $F$ ($\mathbb{R}$ or $\mathbb{C}$) is the most common topology. If your text doesn't say with topology is equiped in $F$, then sure is the euclidean topology.

3.- Yes. $X$ and $F$ are not related in topology. The topology in $X$ can be different from the topology in $F$.

4.- Not in general. If we have $X$ a vector space and consider the indiscrete topology over $X$, then $X$ is not Hausdorff and the sum and product are continuous, i.e., $X$ is a topological vector space.

5.- Is a necessary condition. The definition of a topological vector space is "$X$ is a topological vector space if $X$ is a vector space and the operations of sum and dot product are continuous with the topology of $X$".

6.- Here is a good reference for your question: Topological vector space with discrete topology is the zero space