Definition:Quotient Topological Vector Space
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Definition
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $N$ be a linear subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\tau_N$ be the quotient topology on $X/N$.
We say that $\struct {X/N, \tau_N}$ is the quotient topological vector space of $X$ modulo $N$.
Also see
- Quotient Topological Vector Space is Topological Vector Space verifies that $\struct {X/N, \tau_N}$ is indeed a topological vector space.
- Quotient Topological Vector Space is Hausdorff iff Linear Subspace is Closed shows that $\struct {X/N, \tau_N}$ is Hausdorff if and only if $N$ is closed
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.40$: Definitions