Continuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
Open setIn mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P). More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself.
United Arab EmiratesThe United Arab Emirates (UAE; الإمارات العربية المتحدة al-ʾImārāt al-ʿArabiyya l-Muttaḥida), or simply the Emirates (الإمارات al-ʾImārāt), is a country in West Asia, in the Middle East. It is located at the eastern end of the Arabian Peninsula and shares borders with Oman and Saudi Arabia, while also having maritime borders in the Persian Gulf with Qatar and Iran. Abu Dhabi is the nation's capital, while Dubai, the most populated city, is an international hub.
Interval (mathematics)In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).
South KoreaSouth Korea, officially the Republic of Korea (ROK), is a country in East Asia. It constitutes the southern part of the Korean Peninsula and borders North Korea along the Korean Demilitarized Zone. The country's western border is formed by the Yellow Sea, while its eastern border is defined by the Sea of Japan. South Korea claims to be the sole legitimate government of the entire peninsula and adjacent islands. It has a population of 51.96 million, of which roughly half live in the Seoul Capital Area, the fourth most populous metropolitan area in the world.
Domain (mathematical analysis)In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used for the domain of a function, but in general, functions may be defined on sets that are not topological spaces.
Connected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space is a if it is a connected space when viewed as a subspace of . Some related but stronger conditions are path connected, simply connected, and -connected.
