In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.

Note that the requirement that the maps be continuous is essential ; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton ( a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.

* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.

Context-sensitive grammars are more general than context-free grammars but still orderly enough to be parsed by a linear bounded automaton.

Computationally, a context-sensitive language is equivalent with a linear bounded nondeterministic Turing machine, also called a linear bounded automaton.

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

A linear system that takes an input is called bounded-input bounded-output ( BIBO ) stable if its output will stay bounded for any bounded input.

A bounded linear map, π: A → B, between C *- algebras A and B is called a *- homomorphism if

One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace.

It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are " enough " continuous linear functionals defined on every normed vector space to make the study of the dual space " interesting.

One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C < sub > c </ sub >( X ) extends in exactly one way to a bounded linear functional on C < sub > 0 </ sub >( X ), the latter being the closure of C < sub > c </ sub >( X ) in the supremum norm, and that for this reason the first statement implies the second.

Modern methods include the use of lossless data compression for incremental parsing, prediction suffix tree and string searching by factor oracle algorithm ( basically a factor oracle is a finite state automaton constructed in linear time and space in an incremental fashion ).

In colloquial usage, the terms " Turing complete " or " Turing equivalent " are used to mean that any real-world general-purpose computer or computer language can approximately simulate any other real-world general-purpose computer or computer language, within the bounds of finite memory – they are linear bounded automaton complete.

Olmstead believed Malbolge to be a linear bounded automaton.

In computer science, a linear bounded automaton ( plural linear bounded automata, abbreviated LBA ) is a restricted form of nondeterministic Turing machine.

Since the size of the accessible tape is bounded by some linear function of the length of the input, the linear bounded automaton is computationally equivalent to a nondeterministic Turing machine restricted to the portion of the tape containing the input, hence the name linear bounded automaton.

In 1960, Myhill introduced an automaton model today known as deterministic linear bounded automaton.

Another formalism which is useful to model implementations of hybrid automaton is the lazy linear hybrid automaton.

When this linear draft is completed, I dust it down to a faint image.

The theory predicts a linear dependence of Af on Af, where J is the experimentally determined Curie-Weiss constant.

If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators.

We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

that is, one must know something about D other than the fact that it is a linear operator.

UTC is a discontinuous time scale composed from segments that are linear transformations of atomic time, the discontinuities being arranged so that UTC approximated UT2 until the end of 1971, and UT1 thereafter.

# As exploratory data analysis, an ANOVA is an organization of an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition ( or, equivalently, each set of terms of a linear model ).

# Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors.

Even when the statistical model is nonlinear, it can be approximated by a linear model for which an analysis of variance may be appropriate.

Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach.

ANOVA is considered to be a special case of linear regression

which in turn is a special case of the general linear model.

** On every infinite-dimensional topological vector space there is a discontinuous linear map.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

A linear charge coupled device ( CCD ) array with 200 pixels is used as the detector.

An ideal amplifier would be a totally linear device, but real amplifiers are only linear within limits.

A description of how the device could be used as a shift register and as a linear and area imaging devices was described in this first entry.

This device had input and output circuits and was used to demonstrate its use as a shift register and as a crude eight pixel linear imaging device.

Fairchild's effort, led by ex-Bell researcher Gil Amelio, was the first with commercial devices, and by 1974 had a linear 500-element device and a 2-D 100 x 100 pixel device.

" From these principles and some additional constraints —( 1a ) a lower bound on the linear dimensions of any of the parts, ( 1b ) an upper bound on speed of propagation ( the velocity of light ), ( 2 ) discrete progress of the machine, and ( 3 ) deterministic behavior — he produces a theorem that " What can be calculated by a device satisfying principles I – IV is computable.

Multiple photodiodes may be packaged in a single device, either as a linear array or as a two-dimensional array.

* The amplifying element is biased so the device is always conducting to some extent, normally implying the quiescent ( small-signal ) collector current ( for transistors ; drain current for FETs or anode / plate current for vacuum tubes ) is close to the most linear portion of its transconductance curve.

Non-zero current response is proportional to the voltage supplied and is linear to 60 amperes for this particular ( 25 A ) device.

Amplitude distortion is distortion occurring in a system, subsystem, or device when the output amplitude is not a linear function of the input amplitude under specified conditions.

Nonlinearities in the transfer function of an active device ( such as vacuum tubes, transistors, and operational amplifiers ) are a common source of non-linear distortion ; in passive components ( such as a coaxial cable or optical fiber ), linear distortion can be caused by inhomogeneities, reflections, and so on in the propagation path.

Amplitude distortion is distortion occurring in a system, subsystem, or device when the output amplitude is not a linear function of the input amplitude under specified conditions.

This signal delay will be different for the various frequencies unless the device has the property of being linear phase.

Because group delay is, as defined in ( 1 ), it therefore follows that a constant group delay can be achieved if the transfer function of the device or medium has a linear phase response ( i. e., where the group delay is a constant ).

So if two sine wave signals are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately, with no product terms.

where is the noise factor for the n-th device and is the power gain ( linear, not in dB ) of the n-th device.

Also, calibration should usually be at a number of wavelengths and power levels, since the device is not always linear.

In signal processing, phase distortion or phase-frequency distortion is distortion that occurs when ( a ) a filter's phase response is not linear over the frequency range of interest, that is, the phase shift introduced by a circuit or device is not directly proportional to frequency, or ( b ) the zero-frequency intercept of the phase-frequency characteristic is not 0 or an integral multiple of 2π radians.

This filter reduces the harmonic current, which means that the non-linear device now looks like a linear load.

We assume a “ linear ” device having a transfer function whose small signal form may be expressed in terms of a power series containing only odd terms, making the transfer function an odd function of input signal voltage, i. e., O = − O.