What is space B(X,R)? Let $X = R \ \text{or} \ C$. Let B denote all bounded function from $X$ to $R$.
I am new to the real analysis. I find this concept is somewhat strange to me. 


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*I know $B$ is a set whose elements are functions.Previously, for set, what I met is the set whose element is point in $R^k$. Here, the elements of the set are functions. Can I take those functions as points? How? If not, how to understand functions as elements of set.

*The textbook also call $B$ as space. What's the difference between space and set? I suppose $B$ contains at least three sets, domain and codomain of functions, and functions. In total, they are called space. Am I right? 
If you could provide some intuitive explanations or examples, I will appreciate it.
Edit:
I think we can take the functions as points, if we equip the space with the sup norm. For example, the definition of uniform convergence. 
 A: There is nothing different set-theoretically speaking between sets of points and sets of functions. Just like a set of points, you can ask if a particular function $f:X\to\mathbb{R}$ is included in your set $B$. You can take subsets of $B$, unions of subsets, etc. Don't think of functions and points as being inherently different. Set-theoretically speaking, both are valid elements of sets with the appropriate underlying universe.
For an easier example, consider the functions $f_1, f_2, f_3: \{0,1\} \to \{0,1\}$ given by $f_1(x) = x$, $f_2(x) = (-1)^x$, and $f_3(x) = 0$. Then you can consider the set of functions
$$
A = \{f_1,f_2,f_3\}.
$$
There's nothing else to it, it's the same idea as the set $\{0,1,2\}$ if you don't care about the exact nature of the elements. You can consider $f_1, f_2, f_3$ as "points" of $A$; formally speaking, we should call them elements of $A$, but in analysis it is common to refer to elements as points within certain contexts.
The reason $B$ is referred to as a "space" is because, as Cameron suggests, $B$ satisfies the axioms of a vector space. This is worth verifying as an exercise. A "space" is more generally interpreted as a set with some additional structure, usually of some geometric nature; e.g. metric spaces, vector spaces, topological spaces, etc. The specific type of "space" is generally understood from the context. Analysts sometimes refer to "functions" or "points" as "living" in some sort of "space," but really all this means is that these functions or points lie in some set with some understood structure. We just like to do it because it helps us visualize spaces of functions, which are often infinite-dimensional.
