ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-01

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The quotient by the kernel is isometric to the range with its induced quotient norm

Example

Let $T : X \to Y$ be a bounded linear operator between normed spaces, and write $M : = \ker T$ . The factor map

$\overline{T} : X / M \to im T, \overline{T} (x + M) : = T x,$

is a linear isomorphism. If $im T$ is equipped with the induced quotient norm

$∥ T x ∥_{q u o t} : = ∥ x + M ∥_{X / M},$

then $\overline{T}$ is an isometry.

Facts & Assumptions

Given: A bounded linear operator $T : X \to Y$ and its kernel $M : = \ker T$ .

[L1]

A bounded linear operator is in particular linear (A bounded linear operator between normed spaces).

[L2]

The kernel and image are the sets $\ker T = {x : T x = 0}$ and $im T = {T x : x \in X}$ (Kernel and image of a linear map).

[L3]

The kernel is a linear subspace, and a linear map is injective exactly when its kernel is trivial (The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial).

[L4]

The quotient seminorm is the infimum over representatives (The quotient seminorm (|x+M|{X/M}=\inf{m\in M}|x+m|=\operatorname{dist}(x,M))).

[L5]

A linear map that vanishes on a subspace factors uniquely through the algebraic quotient (Universal property of the quotient vector space).

Verification

technique · direct

1.1

Since $T$ is linear by [L1], the set $M = \ker T$ is a linear subspace by [L3], and certainly $M \subseteq \ker T$ . Therefore [L5] gives a unique linear map $\overline{T} : X / M \to Y$ with $\overline{T} (x + M) = T x$ . Its range is exactly $im T$ , so we may read it as a map into $im T$ .

L1L2L3L5

2.1

If $\overline{T} (x + M) = 0$ , then $T x = 0$ , so $x \in \ker T = M$ by [L2]. Hence $x + M = M$ , the zero coset. Therefore $\overline{T}$ is injective by [L3].

step 1.1L2L3

2.2

If $T x = T y$ , then $x - y \in \ker T = M$ by [L2], so $x + M = y + M$ . Thus the formula $∥ T x ∥_{q u o t} : = ∥ x + M ∥_{X / M}$ is well defined on $im T$ , using the quotient seminorm of [L4]. By construction, $∥ \overline{T} (x + M) ∥_{q u o t} = ∥ x + M ∥_{X / M}$ for every coset, so $\overline{T}$ is an isometry onto $im T$ .

step 1.1L2L4

3.1

Steps 1.1, 2.1, and 2.2 show that $X / \ker T$ is linearly isomorphic to the range of $T$ , and isometric once the range is given the induced quotient norm.

step 1.1step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)