Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm.
Define:
- $A$:={topologies that can make the norm continuous},
- $B$:={topologies that can make the addition and scalar multiplication continuous},
- $C$:={topologies that can make every element in the continuous (wrt $\mathcal{T}$?) dual space $X^{**}$ continuous}. ( The coarsest in $C$ is called the weak topology on $X$?)
Is it true that
- neither of $A$ and $B$ is a subset of the other;
- $A$ can be ordered by finer/coarser, while $B$ cannot be (because the codomain of the norm is $\mathbb{R}$ with a known topology, while the domains and codomains of addition and of scalar multiplication all depend on $X$?);
- $\mathcal{T}$ is the coarsest topology in $A\cap B$;
- $C \equiv A \cap B$.
- Same questions as in 1, 2, and 3, if "norm" is replaced with "inner product" in the above description.
- Are there other types of topologies on a normed/inner product space often encountered, besides those mentioned above?
Thanks and regards!