Dimension Theorem and Rank Theorem
Definition Let $A\in \mathfrak{M}_{m\times n}(\mathbb{R})$ be a matrix. We define the column rank of $A$ as dimension of $\langle [A]^1, \ldots, [A]^n \rangl...
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Closed Sets and Limit Points
Definition (Closed) A subset $A$ of a topological space $X$ is said to be closed if the set $X-A$ is open.
Product Topology and Subspace Topology
Definition (Product Topology on $X \times Y$) Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be topological spaces. The product space topology on $X\times...
Topology and Basis
Definition (Topology) A topology on a set $X$ is a collection $\mathfrak{T}$ having the following properties: (1) $\emptyset$ and $X$ are in $\mathfrak{T}$ (...
Ascoli-Azrela and Stone-Weirstrass Theorem
Theorem (Ascoli-Azrela) Let $\{f_n\}$ be a sequence of functions defined on a set $K$. If $K$ is compact, $\{f_n\}$ is pointwise bounded on $K$, $\{f_n\}$ is...