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Mathematical term
analysis, a Banach limit is a continuous linear functional ϕ : ℓ ∞ → C {\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space ℓ
Banach_limit
Normed vector space that is complete
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
Banach_space
Value approached by a mathematical object
particular value or infinity Banach limit defined on the Banach space ℓ ∞ {\displaystyle \ell ^{\infty }} that extends the usual limits. Convergence of random
Limit_(mathematics)
Theorem on extension of bounded linear functionals
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace
Hahn–Banach_theorem
Theorem about metric spaces
mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem)
Banach_fixed-point_theorem
geometry) Banach fixed-point theorem Banach game Banach lattice Banach limit Banach manifold Banach measure Banach space Banach coordinate space Banach disks
List of things named after Stefan Banach
List_of_things_named_after_Stefan_Banach
Concept in statistics
Cesàro limit of the indicator functions. In cases where the Cesàro limit does not exist this function can actually be defined as the Banach limit of the
Exchangeable_random_variables
Theorem in functional analysis
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball
Banach–Alaoglu_theorem
(x_{n})} is said to be almost convergent to L {\displaystyle L} if each Banach limit assigns the same value L {\displaystyle L} to the sequence ( x n ) {\displaystyle
Almost_convergent_sequence
Theorem stating that pointwise boundedness implies uniform boundedness
boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open
Uniform_boundedness_principle
Mapping function
\lambda } denotes the Lebesgue measure and lim {\displaystyle \lim } the Banach limit. It satisfies 0 ≤ μ ( A ) ≤ 1 {\displaystyle 0\leq \mu (A)\leq 1} and
Sigma-additive_set_function
Property of certain normed spaces
extended Mazur's theorem, which states that the weak limit of a sequence in a Banach space is the limit in the norm of convex combinations of the sequence's
Banach-Saks_property
Instantaneous rate of change (mathematics)
This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is
Derivative
Infinite sum
methods for summing a divergent series are non-constructive and concern Banach limits. A series of real- or complex-valued functions ∑ n = 0 ∞ f n ( x ) {\displaystyle
Series_(mathematics)
Relates three different kinds of weak compactness in a Banach space
three different kinds of weak compactness in a Banach space. Eberlein–Šmulian theorem: If X is a Banach space and A is a subset of X, then the following
Eberlein–Šmulian_theorem
Infinite series that is not convergent
Hahn–Banach theorem that it may be extended to a summation method summing any series with bounded partial sums. This is called the Banach limit. This
Divergent_series
Mathematical concept
specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators.
Approximation_property
Area of mathematics
General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces
Functional_analysis
Metric geometry
are Banach spaces. The space C [ a , b ] {\displaystyle C[a,b]} of continuous real-valued functions on a closed and bounded interval is a Banach space
Complete_metric_space
Mathematics of real numbers and real functions
an example of a Banach space: the metric that this norm defines is complete, which is a consequence of a theorem that the uniform limit of continuous functions
Real_analysis
Generalization of mass, length, area and volume
general, finitely additive measures are connected with notions such as Banach limits, the dual of L ∞ {\displaystyle L^{\infty }} and the Stone–Čech compactification
Measure_(mathematics)
Locally convex topological vector space
linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective
Reflexive_space
Mathematical theorem in real analysis
space. In particular, if Y is a Banach space, then C(X, Y) is itself a Banach space under the uniform norm. The uniform limit theorem also holds if continuity
Uniform_limit_theorem
Concept in mathematics
multidimensional Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions. The Bochner integral provides
Bochner_integral
Mathematical theorem
Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for uniformly convex Banach spaces. It asserts that if K {\displaystyle K} is
Browder_fixed-point_theorem
Use of filters to describe and characterize all basic topological notions and results
Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength
Filters_in_topology
type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure for how far a Banach space is away from
Type and cotype of a Banach space
Type_and_cotype_of_a_Banach_space
Holomorphic functions in infinite dimensions
X is a complex Banach space, is called holomorphic if it is complex-differentiable; that is, for each point z ∈ U the following limit exists: f ′ ( z
Infinite-dimensional holomorphy
Infinite-dimensional_holomorphy
Algebraic trace
the ordinary limit exists limω(α1, α1, α2, α2, α3, ...) = limω(αn) (scale invariance) There are many such extensions (such as a Banach limit of α1, α2,
Dixmier_trace
Generalization of a sequence of points
special type of topological vector space, is a complete TVS (equivalently, a Banach space) if and only if every Cauchy sequence converges to some point (a property
Net_(mathematics)
Generalization of the concept of directional derivative
a complex Banach space X {\displaystyle X} to another complex Banach space Y , {\displaystyle Y,} the Gateaux derivative (where the limit is taken over
Gateaux_derivative
Mathematical function, denoted exp(x) or e^x
function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse e−x
Exponential_function
Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued
Bochner_measurable_function
Mathematical result in the theory of Sobolev spaces
Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful
Aubin–Lions_lemma
Root-finding algorithm
satisfy (at the latest after the first iteration step) the assumptions of the Banach fixed-point theorem. Hence, the error after n steps satisfies | x n − x
Fixed-point_iteration
Theorem in measure theory
by an integrable function then f n → f {\displaystyle f_{n}\to f} in the Banach space L 1 ( S , μ ) {\displaystyle L_{1}(S,\mu )} Without loss of generality
Dominated_convergence_theorem
Type of vector space in math
general Banach spaces. The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to
Hilbert_space
Locally convex topological vector space that is also a complete metric space
generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces
Fréchet_space
Mode of convergence of a function sequence
| , {\displaystyle \|f\|_{\infty }=\sup _{x\in [0,1]}|f(x)|,} this is a Banach space. A sequence f n {\displaystyle f_{n}} in C ( [ 0 , 1 ] ) {\displaystyle
Uniform_convergence
Existence and uniqueness of solutions to initial value problems
follows from the Banach fixed-point theorem that the sequence of "Picard iterates" φ k {\textstyle \varphi _{k}} is convergent and that its limit is a solution
Picard–Lindelöf_theorem
Swedish mathematician and concert pianist
operator is a limit of finite-rank operators. The converse is always true. In a long monograph, Grothendieck proved that if every Banach space had the
Per_Enflo
convergence, and similar to (but distinct from) the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide
Delta-convergence
Theorem
of strongly continuous one-parameter semigroups of linear operators on Banach spaces. It is sometimes stated for the special case of contraction semigroups
Hille–Yosida_theorem
X D , p D ) {\displaystyle \left(X_{D},p_{D}\right)} is a Banach space is called a Banach disk, infracomplete, or a bounded completant in X . {\displaystyle
Auxiliary_normed_space
in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space
Weakly_measurable_function
Space with topology generated by convex sets
of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear
Locally convex topological vector space
Locally_convex_topological_vector_space
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
ISBN 978-87-91180-71-2. Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 978-3-540-52013-9. MR 1102015
Convergence of random variables
Convergence_of_random_variables
Strong form of uniform continuity
special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclusions for
Lipschitz_continuity
S2CID 14254765 Nakamura, Masahiro; Takeda, Ziro (1951), "Group representation and Banach limit", Tôhoku Mathematical Journal, 3 (2): 132–135, doi:10.2748/tmj/1178245513
Uniformly bounded representation
Uniformly_bounded_representation
Concept in mathematical analysis
closed linear subspace of the Banach space of all bounded functions under the uniform distance (hence, it is a Banach space). Pompeiu's above construction
Pompeiu_derivative
Topological space with a dense countable subset
C ( [ 0 , 1 ] ) {\displaystyle C([0,1])} . The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear
Separable_space
space of infinite sequences Das, Gokulananda; Nanda, Sudarsan (2022). Banach limit and applications (1st ed.). Boca Raton: CRC Press. ISBN 978-1-000-46757-4
FK-AK_space
Whose values lie in an infinite-dimensional vector space
infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in most sciences including physics. Set
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Derivative defined on normed spaces
(real or complex) Banach spaces. To do this, let V 1 , … , V n {\displaystyle V_{1},\ldots ,V_{n}} and W {\displaystyle W} be Banach spaces (over the same
Fréchet_derivative
Algebraic structure in linear algebra
function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article, vectors are represented in boldface to distinguish
Vector_space
Statistical parameter
probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends
Concentration_of_measure
Computational tool
continuous dual V ′ of V. Since every vector v in a Banach space V with a Schauder basis is the limit of Pn(v), with Pn of finite rank and uniformly bounded
Schauder_basis
Finest topology making some functions continuous
topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology
Final_topology
Type of topological space
of L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not necessarily the space R {\displaystyle \mathbb {R} }
Bochner_space
a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous. Many standard Banach spaces
Dunford–Pettis_property
On converting relations to functions of several real variables
function theorem in Banach spaces, it is possible to extend the implicit function theorem to Banach space valued mappings. Let X, Y, Z be Banach spaces. Let the
Implicit_function_theorem
Generalization of the exponential function
constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations
C0-semigroup
Relation among continuous functions
the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces
Equicontinuity
Mathematical concept
other algebras. Definitions (1), (2), and (4) all make sense for arbitrary Banach algebras. Some of these definitions require justification to demonstrate
Characterizations of the exponential function
Characterizations_of_the_exponential_function
Function spaces generalizing finite-dimensional p norm spaces
{\displaystyle I} this is a non-separable Banach space which can be seen as the locally convex direct limit of ℓ p {\displaystyle \ell ^{p}} -sequence
Lp_space
Type of topological space
integrals, Banach–Mazur compactum etc. van Douwen, Eric K. (1993). "An anti-Hausdorff Fréchet space in which convergent sequences have unique limits". Topology
Hausdorff_space
Topological space
Several natural commutative Banach algebras are associated with the Cantor space (or group) Δ {\displaystyle \Delta } . The Banach algebra C ( Δ ) {\displaystyle
Cantor_space
finite space Covering space Atlas Limit point Net Filter Ultrafilter Baire category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre
List of general topology topics
List_of_general_topology_topics
Complement of an open subset
When it is not closed, limits of convergent sequences of vectors in the subspace may leave the subspace. Closed subspaces of Banach and Hilbert spaces are
Closed_set
Type of continuous linear operator
operator between Hilbert spaces is the operator-norm limit of finite-rank operators. For general Banach spaces this converse need not hold. The question of
Compact_operator
Mathematical term
convergence and oftentimes viewed weak convergence as preferable. In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous
Weak_topology
Operation in mathematical calculus
generalization of the Lebesgue integral to functions that take values in a Banach space. The collection of Riemann-integrable functions on a closed interval
Integral
Method in mathematical logic
mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method
Fraïssé_limit
is independent of the tags of each partition segment. "On integration in Banach spaces, VI", Ivan Dobrakov and Pedro Morales, Czechoslovak Mathematical
Kolmogorov_integral
Type of mathematical space
topological space; this follows from the Tychonoff theorem. A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is
Compact_space
Theorems connecting continuity to closure of graphs
graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Newton-like root-finding algorithm that does not use derivatives
method naturally generalizes to efficient fixed-point calculation in general Banach spaces, whenever fixed points are guaranteed to exist and fixed-point iteration
Steffensen's_method
Criterion about convergence of series
{\displaystyle \|\cdot \|} is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fréchet derivative.
Weierstrass_M-test
Theorem in mathematics
differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat
Inverse_function_theorem
Property of a sequence or series
convergent series is an equivalent condition for a normed vector space to be Banach (i.e.: complete). Absolute convergence and convergence together imply unconditional
Modes_of_convergence
On strongly convergent combinations of a weakly convergent sequence in a Banach space
\lVert y_{k}-x\rVert \to 0} . For a proof see Ekeland & Temam (1974), p. 6. Banach–Alaoglu theorem – Theorem in functional analysis Bishop–Phelps theorem Eberlein–Šmulian
Mazur's_lemma
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
manifold. All finite-dimensional vector spaces are nuclear. There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice
Nuclear_space
Measure in functional analysis
Lp sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its own right
Lp_sum
Mode of convergence of an infinite series
all of the above holds for integrals with values in a Banach space. The definition of a Banach-valued Riemann integral is an evident modification of the
Absolute_convergence
Generalized object in category theory
monoids, the product is given by the history monoid. In the category of Banach spaces and short maps, the product carries the l∞ norm. A partially ordered
Product_(category_theory)
theorem (functional analysis) Banach–Alaoglu theorem (functional analysis) Banach–Mazur theorem (functional analysis) Banach–Steinhaus theorem (functional
List_of_theorems
Branch of mathematics
equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis. The real numbers provide the standard setting
Mathematical_analysis
Stieltjes measure, however it is often defined for functions with values in a Banach space. It is useful in a number of areas of mathematics, including functional
Laplace–Stieltjes_transform
Locally compact topological group with an invariant averaging operation
German mathematicians use the term "Mittelbare Gruppe") in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation
Amenable_group
Used in the summation of divergent series
large collection of corollaries. The central theorem can now be proved by Banach algebra methods, and contains much, though not all, of the previous theory
Abelian and Tauberian theorems
Abelian_and_Tauberian_theorems
Branch of functional analysis
nest algebras, many commutative subspace lattice algebras, many limit algebras. Banach algebra – Particular kind of algebraic structure Matrix mechanics –
Operator_algebra
Mathematical model of the time dependence of a point in space
necessarily locally homeomorphic to a Banach space, and Φ a continuous function. Being locally homeomorphic to a Banach space allows to use theorems of existence
Dynamical_system
Coarsest topology making certain functions continuous
mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X , {\displaystyle X,} with respect
Initial_topology
Mathematical function with no sudden changes
notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. If f : S → Y {\displaystyle f\colon S\to Y} is not continuous,
Continuous_function
_{X}(\varepsilon ),\quad \varepsilon \in [0,2].} A Banach space is uniformly smooth if and only if the limit lim t → 0 ‖ x + t y ‖ − ‖ x ‖ t {\displaystyle
Uniformly_smooth_space
Mathematical function whose derivative exists
derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is
Differentiable_function
Analogue of a complex analytic space over a nonarchimedean field
k^{n}} whose coordinates have norm at most one. An affinoid algebra is a k-Banach algebra that is isomorphic to a quotient of the Tate algebra by an ideal
Rigid_analytic_space
Function space
{\displaystyle \ell _{p}} the Banach space of all sequences with finite p-norm. Let c 0 {\displaystyle c_{0}} the Banach space of all sequences satisfying
Lorentz_space
Formula for the derivative of a product
\partial x_{3}}\cdot v.\\[-3ex]&\end{aligned}}} Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous
Product_rule
Branch of topology
topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique. Any
General_topology
BANACH LIMIT
BANACH LIMIT
Boy/Male
Hindu
Pearl
Surname or Lastname
English and Welsh
English and Welsh : variant of Bach 3 and 4.
Female
English
English variant spelling of French Blanche, BLANCH means "white."
Girl/Female
Biblical
Who humbles thee, who answers thee.
Male
English
Anglicized form of Hebrew Baruwk, BARUCH means "blessed." In the bible, this is the name of several characters, including a faithful attendant of Jeremiah to whom the apocryphal Book of Baruch is ascribed.
Girl/Female
Arabic
Love
Surname or Lastname
English
English : from Middle English balch, belch ‘balk’, ‘beam’ (Old English bælc, balca), possibly denoting someone who lived in a house with a roof beam rather than in a simple hut; alternatively it may have been a nickname for a man built like a tree trunk, i.e. one of stocky, heavy build.English : nickname from Middle English balche, belche ‘swelling’ (Old English bælc(e)). This was probably chiefly given in the sense ‘swelling pride’, ‘overweening arrogance’, but it can also mean ‘eructation’, ‘belch’ and may therefore in some cases have been acquired by a man given to belching.Welsh : from the adjective balch, which has a range of meanings—‘fine’, ‘splendid’, ‘proud’, ‘arrogant’, ‘glad’—but the predominant meaning is ‘proud’ and from this the family name probably derives.The surname Balch was established in MD c.1650.
Male
English
Anglicized form of Hebrew Chanowk, HANOCH means "dedicated" or "initiated." In the bible, this is the name of the eldest son of Cain, and a son of Jared the father of Methuselah.
Boy/Male
Muslim
Tall and attractive
Boy/Male
Indian
A Hadith was narrated by a Man with the same name
Surname or Lastname
English
English : topographic name for someone who lived by a stream, Middle English beche, Old English bece, a byform of bæce. Compare Bach 3.English : topographic name for someone who lived by a beech tree or beech wood, from Middle English beche ‘beech tree’ (Old English bēce).Perhaps also an Americanized form of German Bisch.John Beach came from England to New Haven, CT, in about 1635. Thomas Beach came from England to Milford, CT, in 1638. It is not clear whether they were related.
Female
Irish
Variant spelling of Irish RÃoghnach, RÃGHNACH means "queen."
Boy/Male
Indian
Tall and attractive
Male
Hebrew
Hebrew name ANATH means "answer (to prayer)." In the bible, this is the name of the father of Shamgar.Â
Male
Irish
Irish name derived from the Gaelic word biorach, BEARACH means "sharp."
Surname or Lastname
English
English : variant of Balch.
Surname or Lastname
English and Irish
English and Irish : variant of Brach 2.
Male
English
Anglicized form of Hebrew unisex Malak, MALACH means "angel, messenger." In the bible, malak is a word used to denote a messenger from God or from a private individual.
Male
Irish
Variant spelling of Irish Bearach, BERACH means "sharp."
Male
Irish
Variant form of Irish Dara, DARACH means "oak."
BANACH LIMIT
BANACH LIMIT
Male
German
Variant spelling of Old High German Walthari, WALTHERE means "ruler of the army."
Female
Hindi/Indian
(लीलावती) Hindi name LEELAVATHI means "free will of God."
Surname or Lastname
English
English : patronymic from Batt 1 or 2.
Boy/Male
Tamil
Bishwa Mohan | பிஷà¯à®µà®¾  மோஹநÂ
Lord Shri Krishna
Girl/Female
Hindu
Boy/Male
Tamil
Devaansh | தேவாஂஷÂ
Part of gods
Boy/Male
Hindu
King of Sun rays
Surname or Lastname
Irish
Irish : variant of Harkin.English and Scottish : patronymic from the personal name Harkin.
Girl/Female
English American Celtic Irish
Abbreviation of Carol and Caroline from the masculine Charles meaning manly.
Girl/Female
Indian, Kashmiri
Full Moon Night
BANACH LIMIT
BANACH LIMIT
BANACH LIMIT
BANACH LIMIT
BANACH LIMIT
v. t.
To place on a bench or seat of honor.
n.
A line of family descent, in distinction from some other line or lines from the same stock; any descendant in such a line; as, the English branch of a family.
a.
Diverging from, or tributary to, a main stock, line, way, theme, etc.; as, a branch vein; a branch road or line; a branch topic; a branch store.
v. t.
A quantity of anything produced at one operation; a group or collection of persons or things of the same kind; as, a batch of letters; the next batch of business.
a.
To bleach by excluding the light, as the stalks or leaves of plants, by earthing them up or tying them together.
n.
The persons who sit as judges; the court; as, the opinion of the full bench. See King's Bench.
v. t.
To form into a bunch or bunches.
n.
A collection, cluster, or tuft, properly of things of the same kind, growing or fastened together; as, a bunch of grapes; a bunch of keys.
a.
To make white by removing the skin of, as by scalding; as, to blanch almonds.
v. t.
To run or drive (as a vessel or a boat) upon a beach; to strand; as, to beach a ship.
v. t.
To cause to turn aside or back; as, to blanch a deer.
n.
Specifically: A breaking or infraction of a law, or of any obligation or tie; violation; non-fulfillment; as, a breach of contract; a breach of promise.
a.
To make white, or whiter; to remove the color, or stains, from; to blanch; to whiten.
a.
To take the color out of, and make white; to bleach; as, to blanch linen; age has blanched his hair.
v. i.
To swell out into a bunch or protuberance; to be protuberant or round.
v. t.
To make a breach or opening in; as, to breach the walls of a city.
n.
A long table at which mechanics and other work; as, a carpenter's bench.
n.
Any division extending like a branch; any arm or part connected with the main body of thing; ramification; as, the branch of an antler; the branch of a chandelier; a branch of a river; a branch of a railway.
n.
A plume or bunch of feathers, esp. such a bunch worn on the helmet; any military plume, or ornamental group of feathers.
n.
To enlarge or dress (a hole), by using a broach.