I'm having a hard time trying to understand what is exactly a bounded set (or subset) in a normed vector space.
This is where I am so far in my "understanding":
- Let $(V, \|\cdot\|)$ be a normed vector space. We can define $\|y-x\| =: d(x,y) =: d$ such that $(V,d)$ is a metric space.
- In a metric space (such as in $(V,d)$), a set $A$ is bounded if it is contained in a ball of finite radius, that is: $A \subset V$ is bounded if there exists $x \in V$ and $0 < R < \infty$ such that $A \subset B(x,R)$, i.e. such that $d(x,y) \le R$ for all $y \in A$.
- A set in a topological vector space (such as a normed vector space) is called bounded if every neighborhood of the zero vector can be inflated to include the set. (Source: Wikipedia)
- Apparently, a subset $A$ of a normed vector space $(V, \|\cdot\|)$ is said to be bounded if there exists $C > 0$ such that $\|v\| \le C$ for all $v \in A$.
What I don't understand is that, to me, it would be "logical" to say that a subset $A$ of a normed vector space $(V, \|\cdot\|)$ is bounded if there exists $x \in V$ and $C>0$ such that $d(x,y) = \|y-x\| \le C$ for all $y \in A$. But it seems like the "real" definition of a bounded (sub)set of a normed vector space is considering only $x=0$, that is, $x$ being the zero vector. Why is that?
Furthermore, I'm not sure to understand the definition given by Wikipedia. Since a neighborhood is a set containing an open set containing a particular point, and considering that this point could be the zero vector, how can we use that information to get the actual definition of a bounded subset of a normed vector space (i.e. the last definition I mentioned in my list above)?
As you can see, I'm quite confused about all this... And I'm a little bit exhausted since I've tried to get it for hours now. Any help would be greatly appreciated...!