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I am struggling with developing an intuition behind 'Affine Subspaces'. So far I have read the theories concerning Affine Subspaces delivered by the course book and visited several websites, however none have made it 100% clear. I feel like I have some sort of intuition for it, but I fail to apply the intuition when it comes to solving problems.

At the moment, I am often required to show why a

*given set*is an

*Affine Subspace*of a

*certain space*. Because I assume it is quite hard for you to convey the intuition behind it without writing a lot, I will try to convey the way I think by approaching a sample problem. I hope you can enlighten me on the errors/misconceptions I make or perhaps add something so I can learn more.

The sample problem:

The sample problem:

*Let X1 = {f : R → R : f(0) = 1 and f is continuous}. Prove that X1 is an affine subspace of C(R, R) (the space of all continuous function with domain R and mapping to R). Hint: You have to do something with the set X0 = {γ : R → R : f(0) = 0 and f is continuous}.*

So, what I think they want me to do at first is two things:

- prove that the set

__X0__is a subspace of

__C(R, R)__(Closed by addition/multiplication)

- define an arbitrary element β which is an element of

__C(R, R)__but not an element of

__X0__

Once I have done both, I feel like I have to define an arbitrary element of X1, for example, the function ƒ, and show that this function contains both an element of X0 (let's say γ) and the arbitrary element β. In other words:

ƒ(x) = γ(x) + β

If this holds then the set X1 is an

*Affine Subspace*of

__C(R, R)__because it contains the sum of a subspace of

__C(R, R)__and an element of

__C(R, R)__?

I feel like I don't fully understand how to construct this function ƒ(x) and which arbitrary β to define. I have scanned through the answer numerous times and I kinda get their reasoning why, but I can't develop my own reasoning. There must be a logical, stepwise approach to the problem.

Thanks in advance!

Erik