Banach spaceIn mathematics, more specifically in functional analysis, a Banach space (pronounced ˈbanax) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.
Quotient space (linear algebra)In linear algebra, the quotient of a vector space by a subspace is a vector space obtained by "collapsing" to zero. The space obtained is called a quotient space and is denoted (read " mod " or " by "). Formally, the construction is as follows. Let be a vector space over a field , and let be a subspace of . We define an equivalence relation on by stating that if . That is, is related to if one can be obtained from the other by adding an element of .
Fréchet spaceIn functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces.
Quotient space (topology)In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its under the canonical projection map is open in the original topological space.
Quotient groupA quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of and defining a group structure that operates on each such class (known as a congruence class) as a single entity.
WhatsAppWhatsApp (also called WhatsApp Messenger) is a freeware, cross-platform, centralized instant messaging (IM) and voice-over-IP (VoIP) service owned by United States tech conglomerate Meta Platforms. It allows users to send text, voice messages and video messages, make voice and video calls, and share images, documents, user locations, and other content. WhatsApp's client application runs on mobile devices, and can be accessed from computers. The service requires a cellular mobile telephone number to sign up.
Direct-attached storageDirect-attached storage (DAS) is digital storage directly attached to the computer accessing it, as opposed to storage accessed over a computer network (i.e. network-attached storage). DAS consists of one or more storage units such as hard drives, solid-state drives, optical disc drives within an external enclosure. The term "DAS" is a retronym to contrast with storage area network (SAN) and network-attached storage (NAS).
IMessageiMessage is an instant messaging service developed by Apple Inc. and launched in 2011. iMessage functions exclusively on Apple platforms: macOS, iOS, iPadOS, and watchOS. Core features of iMessage, available on all supported platforms, include sending text messages, images, videos, and documents; getting delivery and read statuses (read receipts); and end-to-end encryption so only the sender and recipient can read the messages, and no one else, including Apple itself, can read them.
Network-attached storageNetwork-attached storage (NAS) is a file-level (as opposed to block-level storage) computer data storage server connected to a computer network providing data access to a heterogeneous group of clients. The term "NAS" can refer to both the technology and systems involved, or a specialized device built for such functionality (as unlike tangentially related technologies such as local area networks, a NAS device is often a singular unit). A NAS device is optimised for either by its hardware, software, or configuration.
QuotientIn arithmetic, a quotient (from quotiens 'how many times', pronounced ˈkwoʊʃənt) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense, and (a repeating decimal) in the second sense.
Bobbin laceBobbin lace is a lace textile made by braiding and twisting lengths of thread, which are wound on bobbins to manage them. As the work progresses, the weaving is held in place with pins set in a lace pillow, the placement of the pins usually determined by a pattern or pricking pinned on the pillow. Bobbin lace is also known as pillow lace, because it was worked on a pillow, and bone lace, because early bobbins were made of bone or ivory. Bobbin lace is one of the two major categories of handmade laces, the other being needle lace, derived from earlier cutwork and reticella.
