Moore-Penrose Pseudoinverse
Definition Let $V$ and $W$ be finite dimensional inner product spaces over the same field $F$ and let $T:V\to W$ be a linear transformation. Let $L:\ker T^\p...
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Lebesgue Integration
Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...
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...
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...