Is the L2 norm positive definite?

Question

I am learning the concept of $L^2_w[a,b]$ space - space of square-integrable functions on the interval $[a,b]$ - in the context of Hilbert spaces. After struggling to show that the space of continuous functions is not complete, I came to wonder if the $L^2_w$ norm is in fact a well-defined norm.

The definition of a norm, by my understanding, is any function from a vector to a scalar satisfying:

• $|| a || \ge 0$ with $||a||=0$ implying $a=\mathbb{0}$,
• $||\lambda a|| = |\lambda| \cdot ||a||$, and
• $||a+b|| \le ||a|| + ||b||$.

My problem is with the first axiom. It feels like there are plenty of functions in $L^2_w[a,b]$ that are nonzero yet have zero norm. One such example is $f(x)=\begin{cases} 0 & (x \ne \frac{a+b}{2}) \\ 1 & (x = \frac{a+b}{2}) \end{cases}$. These examples seem terrifying to me since, in case of Cauchy sequences in such space, $\lim_{n \rightarrow \infty}{||f_n-f||=0}$ might not imply $\lim_{n\rightarrow\infty}f_n = f$.

Am I missing something or understanding something horribly wrong? Thank you.

Answer 1

The point is the definition of the $L^2$ space: It does NOT consist of square integrable functions but of equivalence classes of square integrable functions. an equivalence class is defined by saying $f \sim g$ iff $f$ and $g$ differ on a set of measure zero.

Answer 2

In $L_\omega^2$ two function are considered identical if they are equal "almost everywhere" - that is, everywhere but on a set of $\omega$ measure $0$ . The actual vectors in that space are the equivalence classes of functions that are equal almost everywhere.