Below you will find example sentences with "locally compact". The examples show how this phrase is used in natural context and which words often surround it.

Locally Compact in a sentence

Corpus data

  • Displayed example sentences: 20
  • Discovered as a combination around: locally
  • Corpus frequency in the collocation scan: 13
  • Phrase length: 2 words
  • Average sentence length: 24.5 words

Words in this phrase

locally compact

Meanings of words in this phrase

  • locally can mean: With respect to place; in place. Frequent word
  • compact can mean: An agreement or contract. Frequent word

Sentence profile

  • Phrase position: 6 start, 11 middle, 3 end
  • Sentence types: 20 statements, 0 questions, 0 exclamations

Usage patterns

These patterns show short text windows around the phrase in the example sentences:

also say locally compact will be 2× a is locally compact 1× a is locally compact if and 1× a locally compact group is 1× and therefore locally compact in senses 1× are not locally compact this is 1× be a locally compact hausdorff space 1× compact is locally compact this condition 1×

Sentence openings and endings

Common sentence openings:

although hausdorff spaces 2× a characterization of 1× a locally compact 1× but in fact 1× commutative c algebras 1× elements of are 1×

Common sentence endings:

compact hausdorff spaces 2× space is preregular 2× a sufficient condition 1× and type iii 1× compact hausdorff space 1× even locally compact 1×

Context words

These words often appear near this phrase in the example sentences:

hausdorff 14× space 10× spaces 10× group 8× type 5× regular 4× non 3× algebra 3× representations 3× lipschitz 2× subset 2× necessary 2×

Related phrases

known locally locally grown locally sourced locally owned

Phrases around this combination

Other phrases discovered around locally:

known locally 9× locally grown 8× locally sourced 8×

Just before it in the phrase index:

loan portfolio loaded onto loaded gun load shedding lloyd wright llc reiterated

Just after it in the phrase index:

locally grown locally owned locally sourced located near locations across lock screen

Other phrases with the same pattern

Same start:

locally grown locally sourced locally owned

Corpus analysis

  • The phrase "locally compact" has 2 words and usually appears in the middle in these examples. The average sentence has 24.5 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as also say locally compact will be, a is locally compact, hausdorff, space and spaces stand out.
  • In the phrase index, this combination connects with known locally, locally grown, locally sourced, known locally, locally grown and locally sourced, linking the page to nearby combinations.

Example types with locally compact

This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:

Short example sentences

Every locally compact Hausdorff space is Tychonoff. (7 words)

Commutative C*-algebras Let X be a locally compact Hausdorff space. (11 words)

Euclidean spaces are locally compact, but infinite-dimensional Banach spaces are not. (12 words)

Longer example sentences

A locally compact group is said to be of type I if and only if its group C*-algebra is type I. However, if a C*-algebra has non-type I representations, then by results of James Glimm it also has representations of type II and type III. (48 words)

Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition. (44 words)

The point at infinity should be thought of as lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea. (34 words)

Example sentences (20)

Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.

However, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact (This condition is sufficient and necessary).

Non-Hausdorff examples * The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3).

Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces.

The first definition is usually taken for locally compact, countably compact, metrisable, separable, countable; the second for locally connected.

The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure (R n with addition is a locally compact group).

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact.

But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point.

Advertentie

The point at infinity should be thought of as lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea.

A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).

A locally compact group is said to be of type I if and only if its group C*-algebra is type I. However, if a C*-algebra has non-type I representations, then by results of James Glimm it also has representations of type II and type III.

Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular.

Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular.

As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.

Commutative C*-algebras Let X be a locally compact Hausdorff space.

Elements of are sometimes called reals as well and though containing a homeomorphic image of in addition to being totally disconnected isn't even locally compact.

Euclidean spaces are locally compact, but infinite-dimensional Banach spaces are not.

Every locally compact Hausdorff space is Tychonoff.

For general non-abelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations.

In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra.

Advertentie