continuous map

[[concept]]

Continuous

Recall that $T : V \to W$ (a map) is continuous on $V$ if for all $v \in V$ and for all sequences ${v_{n}}$ with $v_{n} \to v$ , then $T_{v_{n}} \to T_{v}$ .

Or, equivalently,

For all open $U \subseteq W$ , the set

T^{- 1} (U) = {v \in V | T_{v} \in U} \subseteq V

is open in $V$ .

Mentions

File	Last Modified
absolutely continuous	2025-10-02
Baire Category Theorem	2025-11-09
closed graph theorem	2025-11-09
continuous functions are measurable	2025-11-09
fourier partial sums are given by the Dirichlet kernel	2025-11-09
Functional Analysis Lecture 13	2025-11-09
Functional Analysis Lecture 15	2025-11-09
Functional Analysis Lecture 17	2025-11-09
Functional Analysis Lecture 2	2025-11-09
Functional Analysis Lecture 3	2025-11-09
Functional Analysis Lecture 4	2025-11-09
Functional Analysis Lecture 5	2025-11-09
Functional Analysis Lecture 8	2025-11-09
Functional Analysis Lecture 9	2025-11-09
Hahn-Banach theorem	2025-11-09
Manifold Learning Lecture 08	2025-11-06
Manifold Learning Lecture 12	2025-11-13
Mercer's theorem	2025-10-15
we can always extend functions on subspaces	2025-11-09

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