Intereting Posts

How can I show that $y'=\sqrt{|y|}$ has infinitely many solutions?
Polynomial with infinitely many zeros.
If $G = C_{25}\times C_{45}\times C_{48}\times C_{150}$, where $C_n$ denotes a cyclic group of order $n$, how many elements of order 5 does $G$ have?
Is there a initial “bordism-like” homology theory?
Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.
Sum of Gaussian Sequence
Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?
Can mathematical inductions work for other sets?
Finiteness of the dimension of a normed space and compactness
Is the weak topology sequential on some infinite-dimensional Banach space?
Conditional Expectation of a Poisson Random Variable
Can someone explain the ABC conjecture to me?
How many combinations can be made with these rules? (game of Dobble)
Proving functions are in $L_1(\mu)$.
A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$

How would we prove this? Does this follow from Cauchy Schwarz? Intuitively this is how I see it: In the LHS we could have a negative area that reduces the positive area. In the RHS the area can only increase because we take the absolute values of the functions first.

- How could we solve $x$, in $|x+1|-|1-x|=2$?
- Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.
- How prove this $|x_{p}-y_{q}|>0$
- How is it, that $\sqrt{x^2}$ is not $ x$, but $|x|$?
- How to use triangle inequality to establish Reverse triangle inequality
- for $n$ an integer, why is $n^0=1$ ??

- Almost sure identity $F^{-1}(F(X))=X$ where $F$ is the CDF of $X$
- Some confusion with absolute value
- Prove nth root of n! is less than n+1 th root of ((n+1) !): $\sqrt{n!}\lt \sqrt{(n+1)!}$?
- $|f(x)-f(y)|\le(x-y)^2$ without gaplessness
- Series and integrals for inequalities and approximations to $\log(n)$
- Prove that $y-x < \delta$
- How to solve inequalities with absolute values on both sides?
- Convergence of a recursively defined sequence
- Inequality with condition $x^2+y^2+z^2=1$.
- Integral inequality - does this look correct?

The big idea here is this:

First: it is enough to show that

$$

\left\lvert\int_a^b f(x)\,dx\right\rvert\leq\int_a^b\lvert f(x)\rvert dx,

$$

since you can replace $f(x)$ by $f(x)\cdot g(x)$ to get the desired result.

Now, notice that

$$

-\lvert f(x)\rvert\leq f(x)\leq \lvert f(x)\rvert

$$

for all $x$; hence

$$

-\int_a^b\lvert f(x)\rvert\,dx\leq \int_a^b f(x)\,dx\leq\int_a^b\lvert f(x)\rvert\,dx.

$$

Can you finish it from here?

- Are $C^k$, with the $C^k$ norm, distinct as Banach spaces?
- Sorgenfrey line is hereditarily separable
- Non-isomorphic exact sequences with isomorphic terms
- Why can't Fubini's/Tonelli's theorem for non-negative functions extend to general functions?
- $f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$.
- What are useful tricks for determining whether groups are isomorphic?
- $G\times H\cong G$ with $H$ non-trivial
- If $p$ divides $a^n$, how to prove/disprove that $p^n$ divides $a^n$?
- Definite Integral of $e^{ax+bx^c}$
- How to calculate the index between two complex lattices?
- Permutation without fixed finite set
- Rational quartic curve in $\mathbb P^3$
- When is $R \, A^{-1} \, R^t$ invertible?
- Properties of finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$?
- structure and properties of a function inherited from its integrals