The question stems from my reading of the book Optimization by Vector Space Methods by David Luenberger. To add context, I present the relevant background from the section first:
Section 8.2 Positive Cones and Convex Mappings
By introducing a cone defining the positive vectors in a given space, it is possible to consider inequality (optimization) problems in abstract vector spaces.
Definition. Let $\mathrm{P}$ be a convex cone in a vector space $\mathrm{X}$. For $x, \ y \ \in \mathrm{X}$, we write $x\geq y$ (with respect to $\mathrm{P}$) if $x - y \in \mathrm{P}$. The cone $\mathrm{P}$ defining this relation is called the positive cone in $\mathrm{X}$. The cone $\mathrm{N} = - \mathrm{P}$ is called the negative cone in $\mathrm{X}$ and we write $y\leq x$ for $y- x \in \mathrm{N}$.
For example, in $\mathrm{E}^{n}$, the convex cone $$P = \{x \in \mathrm{E}^{n}: x =(\xi_1, \xi_2, \ldots, \xi_n) ; \xi_i \geq 0 \ \text{for all} \ i\}$$ defines the ordinary positive orthant. In a space of functions defined on an interval of the real line, say $[t_1, t_2]$, it is natural to define the positive cone as consisting of all functions in the space that are nonnegative everywhere on the interval $[t_1, t_2]$.
In the case of a normed space, we write $x>0$ if $x$ is an interior point of the positive cone $\mathrm{P}$. For many applications, it is essential that $\mathrm{P}$ possess an interior point so that the separating hyperplane theorem can be applied. Nevertheless, this is not possible in many common normed spaces. For instance, if $\mathrm{X} = \mathrm{L}_1[t_1, t_2]$ and $\mathrm{P}$ is taken as the subset of nonnegative functions on the interval $[t_1, t_2]$, we can easily show that $\mathrm{P}$ contains no interior point. On the other hand, in $\mathrm{C}[t_1 , t_2]$, the cone of nonnegative functions does possess interior points; for this reason the space $\mathrm{C}[t_1 , t_2]$ is of particular interest for problems involving inequalities.
My Question I have a hard time understanding why the positive cone $\mathbf{P}$ taken as the subset of nonnegative function in the normed space $X=L_1[t_1,t_2]$ contains no interior points. On the other hand, in $C[t_1, t_2]$ the cone of nonnegative functions does possess interior points. Any insights, intuition or example to clarify why this is the case will be highly appreciated. Thanks