TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

If (Y) is Banach then (\mathcal B(X,Y)) is Banach

Statement

Let $X$ and $Y$ be normed spaces over the same scalar field. If $Y$ is Banach, then $B (X, Y)$ is Banach for the operator norm.

Facts & Assumptions

Given: A Banach space $Y$ and an operator-norm Cauchy sequence $(T_{n})$ in $B (X, Y)$ .

[L1]

A Banach space is complete for its norm metric (Banach space).

[L2]

For a bounded operator, the operator norm is the unit-ball supremum and satisfies $∥ T x ∥ \leq ∥ T ∥ ∥ x ∥$ for every $x$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[L3]

$B (X, Y)$ is the vector space of bounded linear operators (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators).

Proof

technique · direct

1.1L1L2choose

Fix $x \in X$ . Since $(T_{n})$ is Cauchy in operator norm, [L2] gives $∥ T_{n} x - T_{m} x ∥ \leq ∥ T_{n} - T_{m} ∥ ∥ x ∥$ , so $(T_{n} x)$ is a Cauchy sequence in $Y$ . Because $Y$ is Banach, there is $T x \in Y$ with $T_{n} x \to T x$ .

2.1step 1.1L3algebra

Step 1.1 defines a map $T : X \to Y$ . If $x, y \in X$ , then $T_{n} (x + y) = T_{n} x + T_{n} y$ for every $n$ , so passing to the limit gives $T (x + y) = T x + T y$ . The same argument with $λ x$ gives $T (λ x) = λ T x$ . Thus $T$ is linear.

2.2step 1.1L2L3

Choose $N$ such that $∥ T_{n} - T_{m} ∥ \leq 1$ for all $m, n \geq N$ . Fix $n \geq N$ and $x \in X$ . For every $m \geq N$ , $∥ T_{m} x ∥ \leq ∥ T_{N} x ∥ + ∥ (T_{m} - T_{N}) x ∥ \leq (∥ T_{N} ∥ + 1) ∥ x ∥$ by [L2]. Letting $m \to \infty$ in step 1.1 gives $∥ T x ∥ \leq (∥ T_{N} ∥ + 1) ∥ x ∥$ , so $T$ is bounded and hence $T \in B (X, Y)$ .

2.3step 1.1L2

Let $ε > 0$ . Since $(T_{n})$ is operator-norm Cauchy, choose $N$ so that $∥ T_{n} - T_{m} ∥ < ε$ for all $m, n \geq N$ . Fix $n \geq N$ and $x$ with $∥ x ∥ \leq 1$ . Step 1.1 gives $T_{m} x \to T x$ , so $∥ (T_{n} - T) x ∥ = \lim_{m} ∥ (T_{n} - T_{m}) x ∥ \leq ε$ by [L2]. Taking the supremum over the unit ball yields $∥ T_{n} - T ∥ \leq ε$ .

3.1step 2.2step 2.3L1∎

Step 2.3 shows $T_{n} \to T$ in operator norm, with $T \in B (X, Y)$ by step 2.2. Therefore every operator-norm Cauchy sequence converges in $B (X, Y)$ , so $B (X, Y)$ is Banach by [L1].

Depends on

Used by

A semigroup with unbounded generator is not norm continuous at zero Corollary
Invertibility and the inverse of the transpose Corollary
A strongly continuous unitary group need not be norm continuous Counterexample
C k map between Banach spaces Definition
Calkin algebra Definition
Complexification and spectrum of a real operator Definition
The negative Sobolev space H⁻¹(Ω) Definition
Bounded operators form a Banach algebra, noncommutative in dimension at least two Example
Spectral projection of an isolated eigenvalue agrees with the riesz projection Example
The square of the Volterra operator has zero trace Example
A positive rank-one Haar average is a nonzero compact intertwiner Lemma
A separably acting abelian von Neumann algebra has a self-adjoint generator Lemma
Bounded Hilbert operators form a C star algebra Lemma
Cauchy estimates for an analytic semigroup give generator power bounds Lemma
Countable compactness closes in the bidual Lemma
James convex-block norm-attainment criterion Lemma
Neumann series and small perturbations of bounded inverses Lemma
Spectral product from traces of powers Lemma
The Dunford contour integral defines a bounded holomorphic family on the sector Lemma
The exponential series of a bounded operator Lemma
Sequence ell-one versus nonatomic L-one for the RNP Remark
The continuous dual, its completeness, and evaluation Remark
Banach closed-range theorem Theorem
Bounded borel pvm integral Theorem
Continuous functional calculus for bounded self adjoint operators Theorem
Hilbert spaces are reflexive by Riesz representation Theorem
Infinite dimensional weak topology is not first countable Theorem
Lumer-Phillips generation theorem Theorem
Minimal C star unitization Theorem
Schauder compact adjoint theorem Theorem
Weakly convergent sequences are norm bounded Theorem

Dependency tree · two levels

8 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)