bounded map on the

USAGE SUMMARY

The phrase "bounded map on the" is correct and usable in written English.
It is typically used in mathematical contexts, particularly in functional analysis or topology, to describe a specific type of function or mapping that has certain boundedness properties. Example: "In our study, we will analyze the properties of a bounded map on the Hilbert space."

✓ Grammatically correct

Science

News & Media

Formal & Business

Human-verified similar examples from authoritative sources

Similar Expressions

60 human-written examples

Let be a strongly positive linear bounded map on with coefficient.

Fixed Point Theory and Applications

We also obtain a new proof of a result of U. Haagerup (Decomposition of completely bounded maps on operation algebras, preprint), characterizing the norm of Schur product maps, and a new Hahn-Banach type extension theorem for these maps.

Journal of Functional Analysis

We use a "Hilbert Schmidt version" of Arveson distance formula to construct an operator space X, isometric to ℓ2⊗ℓ2, such that the space of completely bounded maps on X consists of Hilbert Schmidt perturbations of π(A)⊗Iℓ2.

Journal of Functional Analysis

Under the assumption that the bounded domain D has a C 2 boundary and is strongly pseudo-convex, one can assert that S extends to a bounded mapping on L p ( b D, d σ ), when 1 < p < ∞.

Bulletin of Mathematical Sciences

We introduce the central Haagerup tensor product R ⊗ FhR for a von Neumann algebra R, and we show that the natural injection into tbe space CB R, R) of completely bounded maps on R is isometric.

Journal of Functional Analysis

Let (M,N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property Sσ, and let Φ be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of Φ to a completely bounded mapping on M⊗N.

Journal of Functional Analysis

IfGis a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions onG, is a Banach algebra when represented as an algebra of completely bounded maps on aC*-algebra associated withG.

Journal of Functional Analysis

We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map Mϕ on the reduced free product of the Ai, which acts as a radial multiplier.

Journal of Functional Analysis

We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C∗-algebra of compact operators in terms of tensor products, generalising results of Saar [H.

Journal of Functional Analysis

Proof With the same arguments as given in the first portion of the proof of Theorem 3.3, we know Γ : W 0 → 2 W 0 is a bounded map with convex values and is closed on W 0. Now, we prove the values of Γ are compact in C ( [ 0, 1 ] ; X ).

Boundary Value Problems

Hence we can find uniformly bounded maps h n on each of the intervals, and therefore also uniformly bounded maps g n, which are lower triangular for n in each interval [p j,r j ) with j odd, and upper triangular when j is even.

Complex Analysis and its Synergies

Expert writing Tips

Best practice

When using "bounded map on the", ensure the context clearly defines the domain and codomain of the map, as well as the specific norm or metric used to define boundedness. This is especially important in mathematical writing.

Common error

Avoid assuming that a "bounded map on the" is automatically continuous. While boundedness restricts the range of the map, continuity requires that small changes in the input result in small changes in the output, which are distinct properties. Check the specific properties and be clear in writing.

Linguistic Context

The phrase "bounded map on the" typically functions as a modifier within a mathematical or analytical context. It describes a specific type of mapping, transformation, or function that adheres to certain boundedness criteria. This description is supported by Ludwig AI, which confirms its use in technical contexts.

Expression frequency: Missing

Frequent in

Science

33%

News & Media

33%

Formal & Business

33%

Less common in

Science

0%

News & Media

0%

Formal & Business

0%

Ludwig's WRAP-UP

In summary, the phrase "bounded map on the" is a technical term predominantly used in mathematics and related scientific fields. Ludwig AI confirms that the phrase is grammatically correct and suitable for formal writing. While examples are currently limited, the term's function is to precisely define a type of mathematical mapping based on its boundedness. Related phrases include "bounded operator on the" and "bounded transformation on the". When employing this phrase, ensure clear context and avoid conflating boundedness with continuity.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

bounded mapping on the

Replaces "map" with its synonym "mapping", indicating a transformation or function.

bounded transformation on the

Replaces "map" with "transformation", highlighting the act of changing or altering.

bounded operator on the

Substitutes "map" with "operator", often used interchangeably in functional analysis.

bounded linear map on the

Adds the adjective "linear" to specify the type of transformation being discussed.

limited map on the

Uses "limited" as a synonym for "bounded", indicating restricted range or output.

bounded function on the

Replaces "map" with "function", focusing on the mathematical relationship.

continuous map on the bounded

Changes the order and emphasizes continuity over boundedness.

bounded morphism on the

Uses the term "morphism", which refers to a structure-preserving map.

norm-bounded map on the

Specifies that the boundedness is defined in terms of a norm.

uniformly bounded map on the

Adds the qualifier "uniformly" to specify the type of boundedness.

FAQs

How do you use "bounded map on the" in a sentence?

In functional analysis, you might say, "We analyze the properties of a "bounded map on the" Hilbert space".

What's the difference between a "bounded map on the" and a continuous map?

A "bounded map on the" has a restricted range, while a continuous map ensures small input changes lead to small output changes. They're related but distinct concepts.

Are there alternative terms for "bounded map on the"?

Yes, depending on the context, you can use phrases like "bounded operator on the", "bounded transformation on the", or "bounded function on the".

What does it mean for a map to be 'bounded'?

For a map to be "bounded", it means its output values are limited within a certain range, regardless of the input. This is quantified using norms or metrics in the function's domain and codomain.