Quotient of F-Space by Closed Linear Subspace is F-Space

Theorem

Let $K$ be a topological field.

Let $\struct {X, d}$ be an $F$-space.

Let $N$ be a closed linear subspace of $X$.

Let $X/N$ be the quotient vector space of $X$ modulo $N$.

Let $d_N$ be the quotient metric on $X/N$ induced by $d$.

Then $\struct {X/N, d_N}$ is an $F$-Space.

Proof

Let $\pi : X \to X/N$ be a quotient mapping.

Let $\tau_N$ be the quotient topology of $X$ modulo $N$.

From Quotient Metric on Vector Space induces Quotient Topology, $d_N$ induces $\tau_N$ and $d_N$ is an invariant metric.

From Quotient Topological Vector Space is Topological Vector Space, $\struct {X/N, \tau_N}$ is a topological vector space.

To establish that $\struct {X/N, d_N}$ is an $F$-space, we only need to show that each Cauchy sequence in $\struct {X/N, d_N}$ converges.

Let $\sequence {u_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {X/N, d_N}$.

Pick $n_1 \in \N$ so that:

$\map {d_N} {u_n, u_m} < 2^{-1}$ for all $n, m \ge n_1$.

Now for each $j \ge 2$, inductively pick $n_j > n_{j - 1}$ such that:

$\map {d_N} {u_n, u_m} < 2^{-j}$ for all $n, m \ge n_j$

Then for each $j \in \N$, we have:

$\map {d_N} {u_{n_j}, u_{n_{j + 1} } } < 2^{-j}$

For each $j \in \N$, pick $v_j \in X$ such that $\map \pi {v_j} = u_{n_j}$.

Then for $j \ge 2$, we have:

$\ds \inf_{z \mathop \in N} \map d {v_j - v_{j + 1}, z} < 2^{-j}$

So there exists $z_j \in N$ such that:

$\map d {v_j - v_{j + 1}, z_j} < 2^{-j}$

Then since $d$ is invariant, we have:

$\map d {v_j, v_{j + 1} + z_j} < 2^{-j}$

Set $x_1 = v_1$ and $x_j = v_j + z_{j - 1}$ for $j \ge 2$.

Then, we have:

$\map d {x_j, x_{j + 1} } < 2^{-j}$ for each $j \in \N$

and:

$\map \pi {x_j} = \map \pi {v_j} = u_{n_j}$

from Kernel of Quotient Mapping.

Now, by Metric Space Axiom $(\text M 3)$, we have, for each $m > n$:

$\ds \map d {x_n, x_m} \le \sum_{j \mathop = n}^{m - 1} \map d {x_j, x_{j + 1} } < \sum_{j \mathop = n}^{m - 1} 2^{-j}$

From Sum of Infinite Geometric Sequence, we have:

$\ds \sum_{j \mathop = 0}^\infty 2^{-j}$ converges.

From Cauchy's Convergence Criterion for Series, it follows that for each $\epsilon > 0$ there exists $N \in \N$ such that for $m > n \ge 0$, we have:

$\ds \sum_{j \mathop = n}^{m - 1} 2^{-j} < \epsilon$

so that:

$\map d {x_n, x_m} < \epsilon$

Hence $\sequence {x_n}_{n \mathop \in \N}$ is a Cauchy sequence.

Since $\struct {X, d}$ is an $F$-space, it is in particular a complete metric space.

So $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$.

From the definition of the quotient topology, $\pi$ is continuous.

From Continuous Mapping is Sequentially Continuous, $\pi$ is sequentially continuous.

So $\map \pi {x_j} \to \map \pi x$ as $j \to \infty$.

That is, $u_{n_j} \to \map \pi x$ as $j \to \infty$.

From Convergent Subsequence of Cauchy Sequence, it follows that $\sequence {u_n}_{n \mathop \in \N}$ converges to $\map \pi x$.

Hence every Cauchy sequence in $\struct {X/N, d_N}$ converges.

Hence $\struct {X/N, d_N}$ is a complete metric space.

Hence $\struct {X/N, d_N}$ is an $F$-space.

$\blacksquare$