Intereting Posts

Comparison trees
Understanding the Laplace operator conceptually
Why does $x= \frac{1}{2}(z+\bar{z}) = \frac{1}{2}(z+\frac{r^{2}}{z})$ on the circle?
Control on Conformal map
Every Group is a Fundamental Group
First Cohomology Group
consequence of mean value theorem?
Induction on two integer variables
Pointwise convergence implies $L^p$ convergence?
What are the conditions for integers $D_1$ and $D_2$ so that $\mathbb{Q} \simeq \mathbb{Q}$ as fields.
Where can I find the paper by Shafarevich on the result of the realization of solvable groups as Galois groups over $\mathbb{Q}$?
Nimbers for misère games
Combinatorial proof help
How can I show $U^{\bot \bot}\subseteq \overline{U}$?
Would this proof strategy work for proving the lonely runner conjecture?

I’d like to show that $C([0,1])$ (that is, the set of functions $\{f:[0,1]\rightarrow \mathbb{R} \, \textrm{ and } \, f \, \textrm{is continuous} \}$ is not a complete mertric space under the $L_1$ distance function:

$$

d(f,g) = \int_0^1 |f(x)-g(x)|dx

$$

I can find counter examples (for example, here) but would rather prove it using definitions and principles so that I do not have to rely on committing specific degenerate sequences to memory.

- Inverse of the sum $\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$
- Theorem 6.10 in Baby Rudin: If $f$ is bounded on $$ with only finitely many points of discontinuity at which $\alpha$ is continuous, then
- existence of a special function
- Finding $\lim_{x \to +\infty} \left(1+\frac{\cos x}{2\sqrt{x}}\right)$
- Cauchy's residue theorem with an infinite number of poles
- Show that if $E\subset\mathbb{R}$ is a measurable set, so $f:E\rightarrow \mathbb{R}$ is a measurable function.

Since all compact metric spaces are complete, I have to figure that the place to start is to show that $C([0,1])$ is not compact and that somehow an infinite cover allows for a divergent Cauchy sequence. However, I don’t how to show this (or if it’s even the right approach to take).

- Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$
- Surjective bounded operator in Banach spaces without bounded right-inverse
- $\ell_\infty$ is a Grothendieck space
- About a measurable function in $\mathbb{R}$
- Spectrum of the right shift operator on $\ell^2({\bf Z})$
- Continuation of smooth functions on the bounded domain
- The strong topology on $U(\mathcal H)$ is metrisable
- Tietze extension theorem for complex valued functions
- Norms Induced by Inner Products and the Parallelogram Law
- An other question about Theorem 3.1 from Morse theory by Milnor

$C[0,1]$ can be embedded as a subspace of $L^1[0,1]$. It is dense in $L^1$, but not equal to $L^1$. Therefore it is not closed, and hence not complete.

Since all compact metric spaces are complete, I have to figure that the place to start is to show that $C([0,1])$ is not compact and that somehow an infinite cover allows for a divergent Cauchy sequence.

It is certainly not compact, but the logic is off here. Compact metric spaces are complete, but complete metric spaces need not be compact. For example, think of $\mathbb R$ with its usual metric, or any other nonzero complete normed space.

I will assume you want to show that $C[0,1]$ is not a complete subspace of $L^1$.

Here is a counterexample-free approach.

Suppose $C[0,1]$ is a complete subspace of $L^1[0,1]$, so that it is in particular closed. Since it is clearly a proper subspace, the Hahn-Banach theorem tells us that there is a non-zero continuous linear functional $\phi:L^1[0,1]\to\mathbb R$ such that $\phi$ vanishes on $C[0,1]$. Now the fact that the dual of $L^1[0,1]$ can be identified with $L^\infty[0,1]$ allows us to translate this: there exists a non-zero function $g\in L^\infty[0,1]$ such that $\int_0^1 fg=0$ for all $f\in C[0,1]$.

That this is not possible is a standard result in measure theory.

- Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
- Transitive subgroup of symmetric group $S_n$ containing an $(n-1)$-cycle and a transposition
- Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?
- Inequality $a^2+b^2+c^2 \leq a^2b+b^2c+c^2a+1. $
- Spherical coordinates for sphere with centre $\neq 0$
- Proving the Riemann Hypothesis and Impact on Cryptography
- Is the proof of $\lim_{\theta\to 0} \frac{\sin \theta}{\theta}=1$ in some high school textbooks circular?
- Finding an equation for a circle given its center and a point through which it passes
- Is this equality true or it is not necessarily true?
- Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
- What is the proof that the total number of subsets of a set is $2^n$?
- Which field is it?
- Integral of cosec squared ($\operatorname{cosec}^2x$, $\csc^2x$)
- Conditional Probability and Division by Zero
- Correspondences between Borel algebras and topological spaces