These days I am reading topology.
I came to know about a very beautiful result which says that all the norms on the finite dimensional vector space are equivalent. Using this it was very easy to see that on $GL(n,R)$ all the topologies induced by different norm are equivalent.
Now I am trying to analyze $S=C([a,b])$, set of all continuous real valued function from a compact set $[a,b]$ to $\mathbb{R}$. So far I have seen these three metric in literature.
$X_1: d(f,g)=\sup|f-g|$
$X_2: d(f,g)=\int_{0}^{1}|f-g|dt$
$X_3: d(f,g))=(\int_{0}^{1}|f-g|^2)^{1\over 2}$
Now I want to compare these topologies. I want to deduce which topologies is finer, which is coarser.
To be honest, I don't have any intuition on how to start visualizing open balls in the space.
I would be glad if somebody can show me the path. Thanks in advance.
It's hard to visualize because the vector space $C([a,b])$ is infinite dimensional. But you can use these inequalities:
First $$\int_a^b |f-g|dx \leq (b-a)\sup|f-g|$$
Which means that $\sup|f-g|$ is small implies that $\int_a^b |f-g|dx$ is small.
Similarly $$(\int_a^b |f-g|^2dx )^\frac{1}{2} \leq ((b-a)^2\sup|f-g|^2)^{\frac{1}{2}} = (b-a)\sup|f-g|$$
These two should teach you about the relation between $X_2,X_1$ and $X_3,X_1$. It is left to study the relation between $X_2,X_3$. For this you need a less trivial inequality called the holder inequality. It says that
$$\int_a^b |f(x)-g(x)|dx \leq (b-a)^{1/2}\cdot (\int_a^b |f(x)-g(x)|^{2}dx)^\frac{1}{2}$$
Which means that if the $X_3$'s distance is small then the $X_2$'s distance is small.