∫ab f(x) dx
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Definition (Sobolev Space). The Sobolev space $W^{k,p}(U)$ consists of all locally summable functions $u: U \to \mathbb{R}$ such that for each multi-index $\alpha$ with $|\alpha| \leq k$, the weak derivative $D^{\alpha}u$ exists and belongs to $L^p(U)$.
$$\|u\|_{W^{k,p}(U)} = \left( \sum_{|\alpha| \leq k} \int_U |D^\alpha u|^p \, dx \right)^{1/p}$$
A key tool in the study of Sobolev spaces is the mollifier, which allows approximation of $L^p$ functions by smooth functions.
\begin{definition}[Compact Space]
A topological space $X$ is \textbf{compact} if every
open cover of $X$ has a finite subcover. That is, if
$X = \bigcup_{\alpha} U_{\alpha}$, then there exist
$U_{\alpha_1}, \dots, U_{\alpha_n}$ with
$X = \bigcup_{i=1}^{n} U_{\alpha_i}$.
\end{definition}
\begin{theorem}[Heine–Borel]
A subset $S \subseteq \mathbb{R}^n$ is compact if and
only if it is closed and bounded.
\end{theorem}
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{heine-borel}
\caption{Finite subcover of $[0,1]$}
\end{figure}
Definition (Compact Space). A topological space $X$ is compact if every open cover of $X$ has a finite subcover. That is, if $X = \bigcup_{\alpha} U_{\alpha}$, then there exist $U_{\alpha_1}, \dots, U_{\alpha_n}$ with $X = \bigcup_{i=1}^{n} U_{\alpha_i}$.
Theorem (Heine–Borel)
A subset $S \subseteq \mathbb{R}^n$ is compact if and only if it is closed and bounded.
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Let $\{f_n\}$ be a sequence of measurable functions such that $f_n \to f$ pointwise a.e. If there exists an integrable function $g$ with $|f_n| \leq g$ a.e., then: $$\lim_{n \to \infty} \int f_n \, d\mu = \int f \, d\mu$$
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Add Cauchy–Schwarz proof
Open
Cauchy–Schwarz Inequality.md
+12 −0
1414$$|\langle u, v \rangle| \leq \|u\| \cdot \|v\|$$
1515
16[proof]
17If $v = 0$, the result is trivial. Suppose $v \neq 0$.
18For any $t \in \mathbb{R}$, we have $\|u + tv\|^2 \geq 0$:
19$$0 \leq \|u\|^2 + 2t\langle u,v\rangle + t^2\|v\|^2$$
20Setting $t = -\frac{\langle u,v\rangle}{\|v\|^2}$ yields
21$$|\langle u,v\rangle|^2 \leq \|u\|^2\|v\|^2. \quad\square$$
22[/proof]
1 approval
Fix typo in MVT remark
Open
Mean Value Theorem.md
+1 −1
2828[remark:Geometric Interpretation]
29The MVT says there is always a point on the curve where the tangent
29The MVT says there is always a point on the curve where the **tangent**
3030line is parallel to the secant through $(a, f(a))$ and $(b, f(b))$.
3131This has important consequences for monotonicity: if $f'(x) > 0$
3232on an interval, then $f$ is strictly increasing.
3333[/remark]
2 approvals
Add L'Hôpital's Rule
Merged
L'Hôpital's Rule.md
+8 −0
1[theorem:L'Hôpital's Rule]
2If $\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0$ (or $\pm\infty$),
3and $g'(x) \neq 0$ near $c$, then:
4$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$
5provided the right-hand limit exists.
6[/theorem]
3 approvals
Thread
J
Jake
The derivative of \(e^x\) is itself! So the antiderivative is also \(e^x\). It's the only function with this property.
M
Maria 2:38 PM
That makes sense! So for \(e^{2x}\), it would be \(\frac{1}{2}e^{2x}\) because of chain rule?
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