Some questions of vectors and dense subsets I have a couple of quick functional analysis related questions:
1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. 
Assume further that $K_{m} \subset K$ is dense in $K$. Then if we are given that there exists $G,F \in V^{*}$ such that for some $u_{1} \in K$ we have $$\langle G,v-u_{1} \rangle \geq \langle F,v-u_{1} \rangle  \text{    }\text{    }\text{ for all  } v \in K_{m}$$
then why does it follow immediately that because $K_{m}$ is dense in $K$ we have $$\langle G,v-u_{1} \rangle \geq \langle F,v-u_{1} \rangle \text{    }\text{    }\text{ for all  } v \in K$$ 
How exactly is the density of $K_{m}$ used?
2.Secondly I just wanted to know if when representing vectors as a sketch it is standard to use arrows and points interchangeably? I thought arrow was standard representation. Consider the sketch below from Brezis book where he uses the sketch to show projections on a subset. He uses arrows to show $f-u$ and $v-y$ but uses points for $v,f,u$?  
 A: *

*It is simply about the continuity of $G,F$. If $v\in K$, then there exists a sequence $\{v_n\}$ with $v_n\to v$. Then
$$
\langle G,v-u_1\rangle-\langle F,v-u_1\rangle=\lim_n\langle G,v_n-u_1\rangle-\langle F,v_n-u_1\rangle\geq0.
$$

*It is standard. We do it often in $\mathbb R^2$: if you consider vector operations you will draw them as arrows, but if you are talking say about the continuity of a function $f:\mathbb R^2\to\mathbb R$, you will draw them as points. 
A: *

*I believe separability, completeness and reflexiveness of space and closedness, boundedness and convexity of $K$ are not necessary for this. Fix $v\in K$ and let $\varepsilon > 0$. Since $K_m$ is dense in $K$, there exists $v'\in K_m$ such that $\|v-v'\|<\varepsilon$. We have
$$
\left|\langle G, v\rangle -\langle G, v'\rangle\right|<\|G\|\,\varepsilon 
\implies
\langle G, v\rangle > \langle G, v'\rangle - \|G\|\,\varepsilon
$$
$$
\left|\langle F, v\rangle -\langle F, v'\rangle\right|<\|F\|\,\varepsilon
\implies
\langle F, v'\rangle > \langle F, v\rangle - \|F\|\,\varepsilon
$$
and so
$$
\langle G, v-u_1\rangle =
\langle G, v\rangle - \langle G, u_1\rangle
\geq
\langle G, v'\rangle -\|G\|\,\varepsilon-\langle G, u_1\rangle=
$$
$$
=\langle G, v'-u_1\rangle-\|G\|\,\varepsilon\geq
\langle F, v'-u_1\rangle-\|G\|\,\varepsilon
\geq
$$
$$
\geq\langle F, v-u_1\rangle-\left(\|G\|+\|F\|\right)\varepsilon
$$
Let $\varepsilon\to 0$, it follows that
$$
\langle G, v-u_1\rangle\geq \langle F, v-u_1\rangle
$$

A: (1) $F$,$G$ are continuous and inequality are also conserved when going to the limit. So if your identity is true on a dense susbet, it has a good chance of being true on the whole one.
More precisely, for $v \in K$, there exists a sequence $(v_k) \in K_m^{\mathbb{N}}$ of $K_m$ that goes to $v$. Since $$\langle G, v_k - u_1 \rangle \leq \langle F,v_k - u_2\rangle $$ holds for all $k$, and since $F$ and $G$ are continuous then
$$\langle G, v_k - u_1 \rangle \rightarrow_{k \to \infty} \langle G, v - u_1 \rangle$$
$$\langle F, v_k - u_2 \rangle \rightarrow_{k \to \infty} \langle F, v - u_2 \rangle$$
Since $$\langle G, v_k - u_1 \rangle - \langle F, v_k - u_2 \rangle \leq 0$$
Going to the limit you have that 
$$\langle G, v - u_1 \rangle - \langle F, v - u_2 \rangle \leq 0$$
(2) I would say that the two representations are used and useful, and you can switch from one to another if needed. A point is good to represent a single element, it is the same as an arrow drawn from the origin to this point, and avoid to draw many useless lines, keeping them implicit.
On the other hand, when you start to represent differences, and to compute scalar product, an arrow is much more adapted. For example, if you want to draw as a point the difference between two close points very far away from the origin, such that they are "big" but the difference is "small", then you'all have to go back near the origin, and this is not convenient. When doing scalar product, the angle is also better visualized when drawing the complete arrows. As you can see on the picture, the arrow representing the difference of two vectors drawn as points is just the arrow joining from the first term to the second term, so this is why he used the two ways. If you are not convinced, try to do the same drawing with either only points or arrows, and see that it is not as clear and as clean as this one.
