Some Questions on Topological vector spaces 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 $F$

*

*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)?

*Why is it mentioned in some textbooks that the topology over $F$ is euclidean topology? is this the only topology we can have?

*Does it mean that $F$ can only have euclidean topology while $X$ can have it is own separate and different topology?

*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?

*Does T.V.S imply that addition and multiplication operators are continuous or is it a necessary condition?

*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.
 A: 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
