Is the L2 norm always positive?

Question

Is

$$\int_{a}^{b} u^2(x,t) \, dx \, , \,\, 0\le t \le T$$ defined as the $L^2$-norm:

$$|| u^2||_{2,[a,b]}^{2} $$

Always positive (or equal to zero)? If not what restrictions do I need to make it so? Or does that just depend on the function $u$

Answer 1

The integral of a non-negative function is always non-negative, remember that the integral is monotonic: if $f\leq g$, then $\int f\leq\int g$.

Answer 2

The square of a $\mathbb R$-valued function is never negative, and the integral of a non-negative function is never negative.

When $u : X \to \mathbb C$ then the definition of the norm is changed to $$\|u\|_2^2 = \int \left|u(x)\right|^2 \, dx$$