Quotient space of vector spaces involving continuous functions.

Question

I'm stuck on the follow question:

Let $\mathcal{C}([a,b],\mathbb{R})$ the vector space of the continuous functions $f:[a,b] \to \mathbb{R}$ and let $S$ the subspace of the constant functions of $[a,b]$. Show that the quotient space $\mathcal{C}([a,b],\mathbb{R})/S$ is isomorphic to $W$, where $W$ is the subspace of the continuous functions $g:[a,b] \to \mathbb{R}$ s.t $g(a)=0$.

Usually, this kind of questions is not so hard. I just need some sobrejective linear function $\phi: \mathcal{C}([a,b],\mathbb{R}) \to W$, where $\ker \phi = S$. However, I didn't see this function $\phi$, because this quotient space is kind different. I guess, it is no so intuitive, so do you have some hint for me?

Answer 1

For any $\;f\in \mathcal C(a,b]\;$, define the scalar $\;k(f,a):= f(a)\;$ , and then define

$$\phi: \mathcal C[a,b]\to W\;,\;\;\phi(f):= f(x)-k(f,a)$$

Check stuff...and there you go.

Answer 2

Hint: Try $\phi(f)=x\mapsto f(x)-f(a)$.

Answer 3

$\mathcal{C}([a,b],\mathbb{R})/S$ defined with equivalence $\left(x(t) \sim y(t)\right) \Leftrightarrow \left( x(t) - y(t) = \mathrm{const}\right)$, so $\mathcal{C}([a,b],\mathbb{R})/S$ consists of classes of functions, whose difference is constant (something like in case of antiderivatives). You may think of each class as set of functions $x(t) + C$ for fixed $x(t) \in \mathcal{C}([a,b],\mathbb{R})$. So, in each class you can find function that is zero in $t = a$ and vice versa, for each function $f(t) \in \mathcal{C}([a,b],\mathbb{R})$ with $f(a) = 0$ there is a class of equivalent functions if form of $f(t) + C$. So, this mapping is isomorphism you are looking for.