bounded operator on the

USAGE SUMMARY

The phrase "bounded operator on the" is correct and usable in written English.
It is typically used in mathematical or functional analysis contexts to refer to a specific type of operator that is bounded in a certain space. Example: "In functional analysis, a bounded operator on the Hilbert space is one that maps bounded sets to bounded sets."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

9 human-written examples

Let be a bounded operator on, the one has the following.

Journal of Inequalities and Applications

But is a bounded operator on the set of continuous functions endowed with the uniform norm.

Advances in Difference Equations

It is obvious that each weighted composition operator is a bounded operator on the ball algebra.

Journal of Inequalities and Applications

It is a bounded operator on the set of uniformly bounded functions endowed with the uniform topology.

Advances in Difference Equations

It is a bounded operator on the space of continuous bounded functions endowed with the uniform norm.

Advances in Difference Equations

Assume that B is a strongly positive linear bounded operator on the Hilbert space H with a coefficient γ ¯ > 0 and 0 < ϱ < ∥ B ∥ − 1.

Fixed Point Theory and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

51 human-written examples

Let be a bounded operator on, thene one has the following.

Journal of Inequalities and Applications

Let be two commuting bounded operators on the Banach space.

Journal of Inequalities and Applications

Two extreme possibilities are (1 Sisreflexivein the sense that refS=S, and (2 Sistransitivein the sense that refSincludes all bounded operators on the underlying space.

Journal of Functional Analysis

A result of Rudin implies that bounded operators on the Banach space C [0,ω1]) of continuous functions on the ordinal interval [0,ω1] have a natural representation as [0,ω1]-matricesatrices.

Journal of Functional Analysis

Finally, the bounds (4.22), (4.24), and (4.25), and also their variants where (sigma _{2^j}(mu _q^*;h,f)) is replaced by (sigma _{2^j}(mu _j;h,f)), show that the PBF networks constructed there are bounded operators on the spaces involved.

Bulletin of Mathematical Sciences

Expert writing Tips

Best practice

When writing about operators, ensure you clearly define the space on which the operator is bounded. This adds precision to your mathematical statements.

Common error

Avoid using the phrase "bounded operator on the" without specifying the relevant space (e.g., Hilbert space, Banach space). Always provide the specific space to ensure the statement is mathematically meaningful.

Linguistic Context

The phrase "bounded operator on the" functions as a noun phrase, typically acting as a subject or object within a mathematical or functional analysis context. Ludwig's examples showcase it within mathematical theorems and definitions.

Expression frequency: Common

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Encyclopedias

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Ludwig's WRAP-UP

In summary, "bounded operator on the" is a common noun phrase primarily used in scientific and mathematical writing, particularly in functional analysis. Ludwig AI confirms its grammatical correctness. When using this phrase, remember to clearly specify the underlying space to ensure mathematical precision. While alternatives exist, they often carry subtle differences in meaning or emphasis. The phrase functions to define a mathematical object and describe its behavior within a defined space.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

bounded linear operator on the

Adds the adjective 'linear' to further specify the type of operator.

bounded operator on a

Uses the indefinite article 'a' instead of 'the', indicating a less specific context.

bounded operator acting on the

Replaces 'on' with 'acting on', emphasizing the action of the operator.

operator bounded on the

Changes the word order to emphasize the operator characteristic of being bounded.

linear bounded transformation on the

Employs the term 'transformation' instead of 'operator'.

bounded map on the

Uses 'map' as a synonym for 'operator'.

bounded function on the

Substitutes 'operator' with the more general term 'function'.

the bounded operator over

Preposition "on" changed to the preposition "over".

bounded operator within the

The change involves the preposition "on" that has been changed to "within".

bounded operator in the

The preposition "on" is changed to "in".

FAQs

What is a bounded operator?

In functional analysis, a bounded operator is a linear transformation between normed spaces that maps bounded sets to bounded sets. It's a crucial concept in understanding the behavior of operators in infinite-dimensional spaces.

How do I use "bounded operator on the" in a sentence?

You can use "bounded operator on the" when referring to a specific space on which the operator is defined. For example: "Assume that B is a strongly positive linear bounded operator on the Hilbert space H with a coefficient γ ¯ > 0".

What are some alternatives to "bounded operator on the"?

Alternatives include "bounded linear operator on the", "operator bounded on the", or "bounded map on the", depending on the desired level of formality and specificity.

What properties does a "bounded operator on the" typically exhibit?

A "bounded operator on the" typically preserves certain structural properties of the space it acts upon, such as continuity and stability. These operators are essential for studying solutions to differential equations and other mathematical problems.