For the subset, deciding if it is a subspace or not Question and Solution
In my studies I have the following question. I have also attached the provided solution for Exercise 1b (i)
I don't understand how they were able to come with that definition for the subset. Foremost, I don't understand how to interpret the line
S := {f ∈ X | f (0) = 1 + f(1)} ⊂ X
Please if anyone could explain.
 A: Here is the first subset of $X=\mathbb R^{\mathbb R}$:
$$S=\{f\in\mathbb R^{\mathbb R}:f(0)=1+f(1)\}$$
This is the set of all functions $f:\mathbb R\rightarrow\mathbb R$ which satisfy the given condition. For example, consider the linear function $L(x)=1-x$, which passes through $(0,1)$ and $(1,0)$. You can quickly verify that $L(0)=1+L(1)$. Therefore, $L\in S$.
How to check if $S$ is a subspace? By definition, a subspace of $X$ is a subset such that for all functions $f$ and $g$ in the subset and scalars $\alpha$ and $\beta$, the function $(\alpha f+\beta g)$ is also in the subset.
A: You have hit it right on, in that your primary confusion is that you have no idea how to read the set-theoretic notation described in the exercise. For this type of notational business, it helps if someone who knows how to read such notation dictates it to you in plain terms/English. Here is how I would read $S := \{f \in X\mid f(0) = 1 + f(1)\} \subset X$:
The set $S$ is defined to be ($:=$) the set ($\{\dots\}$) of all $f$ in $X$, the vector space of all real valued functions ($f \in X$) such that ( " | " ) $f(0) = 1 + f(1)$. This set is a subset of $X$. ($S \subset X$)
Remember that since they are real valued functions, you can obviously put restrictions on their values.
It is an exercise to you to figure out what the other sets, as defined, mean in plain language. They are very similar worded to the first one.
Once you figure out exactly what types of elements are in your subset of $X$, then the problem asks for you to figure out whether or not $S$ is a subspace of $X$. You have obviously learned how to do the following. You need to figure out: if I have some $f$ and $g$ in $X$, then is $f + g$ in $X$? What about $cf$, where $c$ is a real number?
