What is the $0$ element in the vector space of real-valued functions? To prove that $U$ is a subspace of $V$, which is a vector space over $\textbf{F}$, we check:

*

*$0\in U$

*$u,w \in U$ implies $u + w\in U$

*$(\lambda \in \textbf{F})\wedge(u \in U)$ implies $\lambda u \in U$
But as an example when we consider $U$ as say a set of functions, then what does it mean for $0$ to be included in $U$?
Is it a $0$ function that maps all values to zero? But then since the function is an ordered pair shouldn't any $0$ element be $(0,0)$ which implies that the domain includes $0$, which might not necessarily be the case?
 A: To check whether given set relative to specified field forms vector field or not. You have first study the algebraic operations of set valid there.
For example, if we consider the vector space of real number over real numbers field exactly the $0\in\mathbf{R}$ is the zero element of real number because it is the only zero real number in set such that
$\quad 0+x=x,\quad \forall\quad x\in\mathbf{R}\quad$ which is identity element with respect to addition simply in set of real number.
But when we are talking about the vector space real valued functions, you should keep in mind the two operation, one addition that is: $\quad (f+g)x=f(x)+g(x)\quad$ and the other with respect to the scalar multiplication with function that is defined to be $\quad (cf)(x)=cf(x).$
Now I am  coming towards your question, what does $\quad 0\in S=\text{real valued functions}\quad$
Simply, you can say it is constant function defined to be $\quad O(x)=0,\quad\forall\quad x\in\mathbf{R}\quad$ because it is the only function in set $\quad S\quad$ such that $\quad (f+O)(x)=f(x)+O(x)=f(x)\quad\forall\quad f(x)\in S.$
Moreover, just In my opinion you have no confusion about vector space but are not differentiating well between the definition of each single function in set $S$ and $O(x)$ to each of them.
A: If $V=P_2(\mathbb{R})$, the vector space of polynomials with real coefficients and degree $\leq 2$, then the zero vector of this vector space is a function, since $V$ is a vector space of functions. The zero vector of $V$ is the function $f_\text{zero}\colon\mathbb{R}\to\mathbb{R}$ defined by $f_\text{zero}(x)=0\in\mathbb{R}$. This '$0$' on the right is the actual real number $0$. Once you are comfortable with this idea, you can write $0$ to mean the function $f_\text{zero}$. These symbols are used because we become comfortable and understand when $0$ means the function $f_\text{zero}$ and when it means the actual real number $0$.
