Inclusions and inequalities $L^2$ and $L^\infty$

Question

If $X=(0,T)$ with $1>T>0$, how do we know that $$ L^\infty(X) \subset L^2(X), $$ Then $$\| f \|_{\infty} \leq \|f \|_2$$ Is it true? Why?

Answer 1

Let $X=(0,1)$ and $f(x)=0$ for $x \in (0,1/2)$ and $x=1$ otherwise. Then $||f||_2=1/2$ but $||f||_\infty=1$.

There are bounds you can create between the two, but they depend on the space.