Intereting Posts

Connections of Geometric Group Theory with other areas of mathematics.
Grandi's series contradiction
What are the rules for equals signs with big-O and little-o?
Injective Holomorphic Functions that are not Conformal?
Does the series $\sum \sin^{(n)}(1)$ converge, where $\sin^{(n)}$ denotes the $n$-fold composition of $\sin$?
Fundamental group of the special orthogonal group SO(n)
Compact operator maps weakly convergent sequences into strongly convergent sequences
The method of proving the equality of integrals by showing they agree within $\epsilon$, for an arbitrary $\epsilon>0$
If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease?
What is the difference between all types of Markov Chains?
Does it make any sense to prove $0.999\ldots=1$?
Is there a rule of integration that corresponds to the quotient rule?
Finite Cartesian Product of Countable sets is countable?
If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
Algebra of matrices — equivalence of norms

Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and $d_{\infty}$ are topologically equivalent in $C^1([0,1])$ (where $d_{\infty}=sup_{x \in [0,1]}|f(x)-g(x)|)$

My attempt at a solution:

If two metrics are topologically equivalent, then they have the same convergent sequences. Honestly, I couldn’t do anything. I am trying to define a sequence of functions $\{f_n\}_{n \in \mathbb N}$ such that $f_n \to f$ in, for instance, $(C^1([0,1]),d)$ but $f_n \not \to f$ in $(C^1([0,1]),d_{\infty})$. Could it be this two metrics are topologically equivalent? If this is the case, how could I prove it? If not, I would appreciate any hint to find an adequate sequence of functions that works for what I am trying to prove.

- Find all continuous functions $f(x)^2=x^2$
- Prob. 3, Chap. 3 in Baby Rudin: If $s_1 = \sqrt{2}$, and $s_{n+1} = \sqrt{2 + \sqrt{s_n}}$, what is the limit of this sequence?
- A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$
- Is this proof of the fundamental theorem of calculus correct?
- Asymptotic expansion of $\int_0^{2\pi}ae^{x\cos a}da$
- Proof explanation of Stone-Weierstrass theorem

- Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$
- A proof of Stirling's Formula
- Understanding of nowhere dense sets
- A function is not continuous, but the image of convergent sequences converge
- What are the consequences if Axiom of Infinity is negated?
- Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?
- Continuous map $f : \mathbb{R}^2\rightarrow \mathbb{R}$
- Topology: reference for “Great Wheel of Compactness”
- Examples of a quotient map not closed and quotient space not Hausdorff
- Integral $\int_0^\pi \cot(x/2)\sin(nx)\,dx$

Set $f_n(x) = \frac{1}{n} \sin(nx)$. It converges to zero in $(C^1([0,1]),d_{\infty})$ but it doesn’t converge to zero in $(C^1([0,1]),d)$, because there is always a point $x$ such that $f’_n(x) = 1$.

- Proving that second derivative is perpendicular to curve
- Difference between logarithm of an expectation value and expectation value of a logarithm
- Every abelian group of finite exponent is isomorphic to a direct sum of finite cyclic groups?
- Consider the set of all $n\times n$ matrices, how many of them are invertible modulo $p$.
- Traditional combination problem with married couples buying seats to a concert.
- number of non-isomorphic rings of order $135$
- What does this music video teach us about 863?
- Prove that $\dfrac{\pi}{\phi^2}<\dfrac{6}5 $
- Any employment for the Varignon parallelogram?
- Show that $\binom{2n}{ n}$ is divisible by 2?
- The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable
- explanation for a combinatorial identity involving the binomial coefficient
- Localisation isomorphic to a quotient of polynomial ring
- Graph of $|x| + |y| = 1$
- Real analysis: density and continuity