Affine Subspace in $\Bbb R^n$ Proof Let  $c$ ∈ $\mathbb{R}$  and $U$ be a subspace of $\mathbb{R^n}$ and $U$ is not empty. Further let $\langle .,.\rangle$ be the standard inner product. 
Show that 

$$U_c:= \{v ∈ \mathbb{R^n}: \forall u ∈ U;\langle v,u\rangle=c\} $$
  is an affine  subspace.

Can someone check if my proof is right ?  

1) Let $u_1$ and $u_2$ be two vectors in $\mathbb{R^n}$,then <$v$,$u_1$+$u_2$>= $c$ ,  
<$v$, $u_1$> + <$v$,$u_2$>  ∈ $U_c$
2) Let $a$ be a scalar  ∈ $\mathbb{R}$, then  <$a$$v$,$u$>=$a$.<$v$,$u$>= $a$$c$ ∈ $U_c$ 

So $U_c$ is an affine subspace 
 A: Your proposed proof is flawed in ways that John Hughes has pointed out, but also more fundamentally because you seem to want to apply the definition of vector subspace, not that of affine subspace. Affine subspaces are not define by closure under addition and scalar multiplication; indeed unless they happen to also be vector subspaces (namely when they pass through the origin) they do not have these closure properties.
For an actual proof you could start observing that
$$
  U_c=\{\,v ∈ \mathbb R ^n\mid \forall u \in U:\langle v,u\rangle=c\,\}
 =\bigcap_{u\in U}\{\,v ∈ \mathbb R ^n\mid \langle v,u\rangle=c\,\},
$$
and since the intersection of affine subspaces always gives an affine subspace (I am assuming your definition of affine subspace allows for the empty set, as per my comments under the question; in any case the result is not true if one assumes the contrary), it suffices to show that each set $\{\,v ∈ \mathbb R ^n\mid \langle v,u\rangle=c\,\}$ is an affine subspace. But that is the solution set of a single linear equation, which should be a standard example of an affine subspace. (The notion of affine subspace is tailored to having the solution sets of systems of linear equations be affine subspaces.) A detailed proof would depend on how they chose to define affine subspaces, which you did not tell us.
By the way, this proof shows that any hypothesis about $U$, beyond that it is a subset of $\Bbb R^n$, is superfluous. Also it is weird to require all equations to have the same right hand side $c$ (one could have a different $c$ for each $u$, this makes no difference in the proof), but it does not actually hurt.
A: Welcome to MSE. You've written an attempted proof, but it has many errors. I'll point them out, and then suggest an approach you might take to create a correct proof. 

1) Let $u_1$ and $u_2$ be two vectors in $\mathbb{R^n}$,then <$v$,$u_1$+$u_2$>= $c$ ,  

How can you conclude, merely from knowing that $u_1$ and $u_2$ are any vectors in $\Bbb R^n$, that the inner product of $u_1 + u_2$ with $v$ is equal to $c$? You can't conclude that, because it's not true. Consider the case, in $\Bbb R^2$, where $v = \pmatrix{1\\0}$ and $c = 0$. Let $u_1 = u_2 = v$. Then the inner product is $2$, but $c$ is zero, so your claim is false. 

<$v$, $u_1$> + <$v$,$u_2$>  ∈ $U_c$

This statement doesn't even make sense: the two items on the left are real numbers, whose sum is a real number. $U_c$ is a subspace of $\Bbb R^n$, so unless $n = 1$, it makes no sense to say that a number is in $U_c$. 

2) Let $a$ be a scalar  ∈ $\mathbb{R}$, then  <$a$$v$,$u$>=$a$.<$v$,$u$>= $a$$c$ ∈ $U_c$ 

Same deal: the left hand side is a real number (the product of $a$ and $c$), while the right hand side is a set of vectors. 

An approach you might want to take for fixing all this (or "replacing", more likely). 


*

*Write down the definition of what it means for $U_c$ to be an affine subspace. This amounts to simply copying the definition from your text or notes, but substituting $U_c$ for whatever letter was used to denote the affine subspace. 

*To be an affine subspace, $U_c$ will have to satisfy some properties. Prove these one at a time. To write your proof, try using "two column" format, where you give a statement and a reason at every step. You might start out with something like this to prove that a set $H$ is a subspace:
S1. To prove $H$ is a subspace, I need to show that if $a, b \in H$, then $a + b \in H$. 
R1. The is the first part of the definition of subspace. 
S2. Suppose that $a, b \in H$. 
R2. Hypothesis for an if-then proof. 
S3. ...
Continue in this manner, filling in a reason for every statement you make, until you arrive at the desired conclusion. 
Best of luck. 
