Definition:Quotient Topological Vector Space

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