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6 notes · Last updated May 23, 2026
6
Total Notes
5
Theorems
0
Proofs
1
Definitions
4
Topics
Notes
ANALYSIS
Cauchy Criterion for Convergence
A sequence of real numbers converges if and only if it is a Cauchy sequence — providing convergence without knowing the limit.
\forall\varepsilon>0\;\exists N\in\mathbb{N}\;\forall m,n>N:\;|a_n-a_m|<\varepsilon
2026-05-18 Theorem
ALGEBRA
Rank-Nullity Theorem
For a linear map between finite-dimensional vector spaces, dim(V) = rank(T) + nullity(T).
\dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)
2026-05-15 Theorem
ANALYSIS
Heine-Cantor Theorem
A continuous function on a compact metric space is uniformly continuous.
\forall\varepsilon>0\;\exists\delta>0\;\forall x,y\in K:\;d(x,y)<\delta\Rightarrow d(f(x),f(y))<\varepsilon
2026-05-12 Theorem
PROBABILITY
Central Limit Theorem
The normalised sum of i.i.d. random variables with finite variance converges in distribution to the standard normal.
\sqrt{n}\,\frac{\bar{X}_n-\mu}{\sigma}\xrightarrow{\mathcal{D}}\mathcal{N}(0,1)
2026-05-10 Theorem
TOPOLOGY
Compact Subsets of $\mathbb{R}^n$
A subset K of ℝⁿ is compact if and only if it is both closed and bounded (Heine-Borel).
K\text{ compact}\iff K\text{ closed}\land K\text{ bounded}
2026-05-08 Definition
ALGEBRA
Spectral Theorem
Every real symmetric matrix is orthogonally diagonalisable with real eigenvalues.
A=A^\top\implies A=QDQ^\top,\quad Q\text{ orthogonal}
2026-05-05 Theorem
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