Spectrum of Multiplication Operator $T$ in $C[0,1]$ "Let $X=C[0,1]$ and $v \in X$ be a fixed function. Let $T$ be the multiplication operator by $v$, i.e. $Tx(t)=v(t)x(t)$. Find the spectrum of $T$."
This is an exercise from a PDF of notes I found online. I'm trying to better understand Spectral Theory.
So $\lambda$ is a regular value if $(T-\lambda I)^{-1}$ exists, is bounded, and dense (I think there is a lemma which lets us not worry about the dense part). The set of all regular values is $\rho(T)$ and the spectrum is $\sigma(T)=\mathbb{C}\setminus \rho(T)$.
It seems to me that $(T-\lambda I)^{-1}$ maps some $y(t)$ to $\frac{y(t) + \lambda}{v(t)}$. However, this would be problematic if $v(t)=0$ for some values of of $t \in [0,1]$
I don't have much of an understanding of all these definitions so if someone could give a solution to this example, I think that would help clear up some of the ideas for me. 
Thanks in advance.

Variation of T.A.E.'s answer to: $\mathbb{C}\setminus\mathcal{R}(v)\subseteq\rho(T)$
Suppose $\lambda\notin\mathcal{R}(v)$.
Regard $w:=\frac{1}{v-\lambda}$.
Then it exists and it is continuous. Moreover, it is bounded as it lives on a compact set. 
Thus, its multiplication is an everywhere welldefined and bounded operator:
$$\|M_w g\|_\infty\leq\|w\|_\infty\cdot\|g\|_\infty,\quad g\in\mathcal{C}([0,1])$$
Especially, its multiplication is the inverse:
$$M_w(M_v-\lambda)f=\frac{1}{v-\lambda}(v-\lambda)f=f$$
$$(M_v-\lambda)M_wg=(v-\lambda)\frac{1}{v-\lambda}g=g$$
Concluding $\lambda\in\rho(T)$.
