For $[a,b] \subseteq \mathbb{R}$, consider the set $C([a,b]):=\{f:[a,b] \rightarrow \mathbb{R}: $f$ \text{ is continuous} \}$. Define $||f||_p:=\biggl(\int_a^b |f(x)|^p dx\biggr)^{\frac{1}{p}}$
a) Is $C([a,b])$ a vector space (over $\mathbb{R} ,\mathbb{C}$)?
b) Is $(C([a,b]),||\cdot||_p)$ a normed vector space?
What I did:
Regarding a)
By pointwise evaluation, it is easy to see that $C([a,b])$ is a vector space over $\mathbb{R}$ and over $\mathbb{C}$, in the complex case one just has to split everything into real and imaginary parts.
Regarding b)
For $||\cdot||_p$ to be a norm we have to show that $||f||_p=0 \Rightarrow f=0$.
Since $f$ is continuous, we can define $g(x):=|f(x)|^p$ which is also continuous. the fundamental theorem of calculus ensures us that $g$ has an antiderivative $G(x):= \int_a^x g(t) dt$.
But since $g(x) \geq 0$, the function $G$ is monotonically increasing. Hence, $G(a) \leq G(b)$. But this implies that $\int_a^b g(x) dx =G(b) - G(a) \geq 0$.
If now $\int_a^b g(x) dx=0$, this implies that G(b)-G(a)=0
Because $G$ is monotonically increasing, $G$ has to be constant. Thus its derivative is 0, i.e. $g=0$.
Further $|| \lambda f ||_p= |\lambda| ||f||_p$. And by the Minkowski inequality we see that $||f+g||_p \leq ||f||_p+||g||_p$.
So my answer is, that $(C[a,b], ||\cdot||)$ is a normed vector space. Is this correct?