I have a more general question about showing that a normed vector space is not complete with respect to a given norm. We know that $C([0,1])$ is complete with respect to the sup norm. To show that $C([0,1])$ is not complete with respect to $L^1$-norm, one can exhibit an $L^1$-Cauchy sequence $f_n \in C([0,1])$ and show that it cannot converge to a continuous function with respect to the $L^1$-norm. However, does it also suffice to show that the sup norm and $L^1$-norm are not equivalent in this space? Consider the sequence of functions $f_n(x) = x^n$. Then $$\|f_n\|_{\infty} = 1 \mbox{ for all } n \in \mathbb{N}$$ while $$\|f_n\|_{L^1} = \int_0^1 x^n dx = \frac{1}{n+1} \rightarrow 0 \mbox{ as } n \rightarrow \infty$$ so there does not exist a $C > 0$: $$\|f_n\|_{\infty} \leq C \|f_n\|_{L^1} \mbox{ for all } n \in \mathbb{N}.$$ Since these norms are not equivalent and $C([0,1])$ is complete with respect to the sup norm, does it necessarily follow that $C([0,1])$ is not complete with respect to the $L^1$-norm?
C([0,1]) Not Complete with Respect to L^1-Norm
complete-spacesreal-analysis
Best Answer
No, that would not be a valid proof. As you have stated, two norms are equivalent iff they have the same Cauchy sequences and thus, if one norm is complete, any equivalent norm is also complete. However, two non-equivalent norms can both make a space complete. This is because non-equivalent norms have different Cauchy sequences and difference convergent sequences.
As an example, let $(X, \|\cdot \|_1)$ be a complete normed space and take $f:X \to \mathbb C$ to be an unbounded linear functional. Let $y \in X$ be such that $f(y) = 1$ (we can always find such $y$ by scaling $f$ if necessary). Next, define $T : X \to X$ by $T(x) = x - 2f(x)y$ for $x \in X$. Then $$T^2(x) = x-2f(x)y - 2f(x-2f(x)y)y = x - 2f(x)y - 2f(x)y + 4f(x)y =x$$ so $T^2 = I$. Thus, $$\| x \|_2 := \|T(x) \|_1, \,\,\,\, x \in X$$ defines a new norm which is not equivalent to $\| \cdot \|_1$ but which is still complete.
This construction is discussed here: https://www.researchgate.net/publication/226200984_Equivalent_complete_norms_and_positivity