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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, a frame of an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
is a generalization of a
basis of a vector space In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
to sets that may be
linearly dependent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. In the terminology of
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, a frame provides a redundant, stable way of representing a
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
. Frames are used in
error detection and correction In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communi ...
and the design and analysis of
filter bank In signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, each one carrying a single frequency Sub-band coding, sub-band of the original signal. One application of ...
s and more generally in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
.


Definition and motivation


Motivating example: computing a basis from a linearly dependent set

Suppose we have a set of vectors \ in the vector space ''V'' and we want to express an arbitrary element \mathbf \in V as a linear combination of the vectors \, that is, we want to find coefficients c_k such that : \mathbf = \sum_k c_k \mathbf_k If the set \ does not span V, then such coefficients do not exist for every such \mathbf. If \ spans V and also is
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, this set forms a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
of V, and the coefficients c_ are uniquely determined by \mathbf. If, however, \ spans V but is not linearly independent, the question of how to determine the coefficients becomes less apparent, in particular if V is of infinite dimension. Given that \ spans V and is linearly dependent, one strategy is to remove vectors from the set until it becomes linearly independent and forms a basis. There are some problems with this plan: # Removing arbitrary vectors from the set may cause it to be unable to span V before it becomes linearly independent. # Even if it is possible to devise a specific way to remove vectors from the set until it becomes a basis, this approach may become unfeasible in practice if the set is large or infinite. # In some applications, it may be an advantage to use more vectors than necessary to represent \mathbf. This means that we want to find the coefficients c_k without removing elements in \. The coefficients c_k will no longer be uniquely determined by \mathbf. Therefore, the vector \mathbf can be represented as a linear combination of \ in more than one way.


Formal definition

Let ''V'' be an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
and \_ be a set of vectors in V. These vectors satisfy the ''frame condition'' if there are positive real numbers ''A'' and ''B'' such that 0 and for each \mathbf in ''V'', : A \left\, \mathbf \right\, ^2 \leq \sum_ \left, \langle \mathbf, \mathbf_k \rangle \ ^2 \leq B \left\, \mathbf \right\, ^2 . A set of vectors that satisfies the frame condition is a ''frame'' for the vector space. The numbers ''A'' and ''B'' are called the lower and upper ''frame bounds'', respectively. The frame bounds are not unique because numbers less than ''A'' and greater than ''B'' are also valid frame bounds. The ''optimal lower bound'' is the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of all lower bounds and the ''optimal upper bound'' is the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of all upper bounds. A frame is called ''overcomplete'' (or ''redundant'') if it is not a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for the vector space.


Analysis operator

The operator mapping \mathbf \in V to a sequence of coefficients c_k is called the ''analysis operator'' of the frame. It is defined by: : \mathbf: V \to \ell^2, \quad \mathbf \mapsto \_, \quad c_k = \langle \mathbf,\mathbf\rangle By using this definition we may rewrite the frame condition as : A \left\, \mathbf \right\, ^2 \leq \left\, \mathbf \mathbf \right\, ^2 \leq B \left\, \mathbf \right\, ^2 where the left and right norms denote the norm in V and the middle norm is the \ell ^2 norm.


Synthesis operator

The
adjoint operator In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
\mathbf^* of the analysis operator is called the ''synthesis operator'' of the frame. : \mathbf^*: \ell^2 \to V, \quad \_ \mapsto \mathbf, \quad \mathbf = \sum_k c_k\mathbf_k


Motivation for the lower frame bound

We want that any vector v \in V can be reconstructed from the coefficients \_ . This is satisfied if there exists a constant A > 0 such that for all x,y \in V we have: : A \, x-y \, ^2 \le \, Tx-Ty \, ^2 By setting v=x-y and applying the linearity of the analysis operator we get that this condition is equivalent to: : A \, v \, ^2 \le \, Tv \, ^2 for all v \in V which is exactly the lower frame bound condition.


History

Because of the various mathematical components surrounding frames, frame theory has roots in harmonic and functional analysis,
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, and
matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...
. The
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
has been used for over a century as a way of decomposing and expanding signals. However, the Fourier transform masks key information regarding the moment of emission and the duration of a signal. In 1946,
Dennis Gabor Dennis Gabor ( ; hu, Gábor Dénes, ; 5 June 1900 – 9 February 1979) was a Hungarian-British electrical engineer and physicist, most notable for inventing holography, for which he later received the 1971 Nobel Prize in Physics. He obtaine ...
was able to solve this using a technique that simultaneously reduced noise, provided resiliency, and created quantization while encapsulating important signal characteristics. This discovery marked the first concerted effort towards frame theory. The frame condition was first described by
Richard Duffin Richard James Duffin (1909 – October 29, 1996) was an American physicist, known for his contributions to electrical transmission theory and to the development of geometric programming and other areas within operations research. Education and ...
and
Albert Charles Schaeffer Albert Charles Schaeffer (13 August 1907, Belvidere, Illinois – 2 February 1957) was an American mathematician who worked on complex analysis. Biography Schaeffer was the son of Albert John and Mary Plane Schaeffer (née Herrick). He studied ...
in a 1952 article on nonharmonic
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set (in their terminology, a "
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
frame"). In the 1980s,
Stéphane Mallat Stéphane Georges Mallat (born 24 October 1962) is a French applied mathematician, concurrently appointed as Professor at Collège de France and École normale supérieure. He made fundamental contributions to the development of wavelet theory in ...
,
Ingrid Daubechies Baroness Ingrid Daubechies ( ; ; born 17 August 1954) is a Belgian physicist and mathematician. She is best known for her work with wavelets in image compression. Daubechies is recognized for her study of the mathematical methods that enhance i ...
, and
Yves Meyer Yves F. Meyer (; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017. Biography Born in Paris to a Jewish family, Yves Meyer s ...
used frames to analyze
wavelets A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
. Today frames are associated with wavelets, signal and
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
, and
data compression In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression ...
.


Relation to bases

A frame satisfies a generalization of
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
, namely the frame condition, while still maintaining norm equivalence between a signal and its sequence of coefficients. If the set \ is a frame of ''V'', it spans ''V''. Otherwise there would exist at least one non-zero \mathbf \in V which would be orthogonal to all \mathbf_k. If we insert \mathbf into the frame condition, we obtain : A \left\, \mathbf \right\, ^2 \leq 0 \leq B \left\, \mathbf \right\, ^ ; therefore A \leq 0, which is a violation of the initial assumptions on the lower frame bound. If a set of vectors spans ''V'', this is not a sufficient condition for calling the set a frame. As an example, consider V = \mathbb^2 with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
, and the infinite set \ given by :\left\. This set spans ''V'' but since \sum_k \left, \langle \mathbf_k , (0,1)\rangle \ ^2 = 0 + 1 + \tfrac + \tfrac +\dotsb = \infty, we cannot choose a finite upper frame bound ''B''. Consequently, the set \ is not a frame.


Applications

In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, each vector is interpreted as a signal. In this interpretation, a vector expressed as a linear combination of the frame vectors is a redundant signal. Using a frame, it is possible to create a simpler, more sparse representation of a signal as compared with a family of elementary signals (that is, representing a signal strictly with a set of linearly independent vectors may not always be the most compact form). Frames, therefore, provide ''robustness''. Because they provide a way of producing the same vector within a space, signals can be encoded in various ways. This facilitates
fault tolerance Fault tolerance is the property that enables a system to continue operating properly in the event of the failure of one or more faults within some of its components. If its operating quality decreases at all, the decrease is proportional to the ...
and resilience to a loss of signal. Finally, redundancy can be used to mitigate
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
, which is relevant to the restoration, enhancement, and reconstruction of signals. In signal processing, it is common to assume the vector space is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
.


Special cases


Tight frames

A frame is a ''tight frame'' if ''A'' = ''B''; in other words, the frame satisfies a generalized version of
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
. For example, the union of ''k'' disjoint
orthonormal bases In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of a vector space is a tight frame with ''A'' = ''B'' = ''k''. A tight frame is a ''Parseval frame'' (sometimes called a ''normalized frame'') if ''A'' = ''B'' = 1. Each orthonormal basis is a Parseval frame, but the converse is not always true. A frame \_ for V is tight with frame bound ''A'' if and only if : v = \frac \sum_k \langle \mathbf,\mathbf_k\rangle \mathbf_k for all v \in V.


Equal norm frame

A frame is an ''equal norm frame'' (sometimes called a ''uniform frame'' or a ''normalized frame'') if there is a constant ''c'' such that \, e_i\, = c for each ''i''. An equal norm frame is a ''unit norm frame'' if ''c'' = 1. A Parseval (or tight) unit norm frame is an orthonormal basis; such a frame satisfies
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
.


Equiangular frames

A frame is an ''equiangular frame'' if there is a constant ''c'' such that , \langle e_i, e_j \rangle , = c for each distinct ''i'' and ''j''.


Exact frames

A frame is an ''exact frame'' if no proper subset of the frame spans the inner product space. Each basis for an inner product space is an exact frame for the space (so a basis is a special case of a frame).


Generalizations

A '' Bessel Sequence'' is a set of vectors that satisfies only the upper bound of the frame condition.


Continuous frame

Suppose ''H'' is a Hilbert space, X a
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
, and \mu is a locally finite ''
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...
'' on X. Then a set of vectors in ''H'', \_ with a measure \mu is said to be a ''Continuous Frame'' if there exists constants, 0 such that A, , f, , ^2\leq \int_, \langle f,f_x\rangle, ^2d\mu(x)\leq B, , f, , ^2 for all f\in H.


Example

Given a discrete set \Lambda\subset X and a measure \mu= \delta_\Lambda where \delta_\Lambda is the ''
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
'' then the continuous frame property :A, , f, , ^2\leq \int_, \langle f,f_x\rangle, ^2d\mu(x)\leq B, , f, , ^2 reduces to :A, , f, , ^2\leq \sum_, \langle f,f_\rangle, ^2\leq B, , f, , ^2. and we see that continuous frames are indeed the natural generalization of the frames mentioned above. Just like in the discrete case we can define the analysis, synthesis, and frame operators when dealing with continuous frames.


Continuous analysis operator

Given a continuous frame \_ the ''continuous analysis operator'' is the operator mapping \_ to a sequence of coefficients \langle f,f_x\rangle _. It is defined as follows: : T:H \mapsto L^2(X,\mu) by f \to \langle f,f_x\rangle _.


Continuous synthesis operator

The adjoint operator of the continuous analysis operator is the ''continuous synthesis operator'', which is the map : T^*:L^2(X,\mu) \mapsto H by a_x \to\int_X a_x f_x d\mu(x).


Continuous frame operator

The composition of the continuous analysis operator and the continuous synthesis operator is known as the ''continuous frame operator''. For a continuous frame \_, it is defined as follows: : S:H\mapsto H by Sf:=\int_X \langle f,f_x\rangle f_x d\mu(x).


Continuous dual frame

Given a continuous frame \_, and another continuous frame \_, then \_ is said to be a ''continuous dual frame'' of \ if it satisfies the following condition for all f, h\in H: : \langle f, h\rangle =\int_X\langle f,f_x\rangle \langle g_x, h\rangle d\mu(x).


Dual frames

The frame condition entails the existence of a set of ''dual frame vectors'' \ with the property that : \mathbf = \sum_k \langle \mathbf , \mathbf_k \rangle \mathbf_k = \sum_k \langle \mathbf , \mathbf_k \rangle \mathbf_k for any \mathbf \in V. This implies that a frame together with its dual frame has the same property as a basis and its
dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with th ...
in terms of reconstructing a vector from scalar products. In order to construct a dual frame, we first need the linear mapping \mathbf : V \rightarrow V, called the frame operator, defined as : \mathbf \mathbf = \sum_ \langle \mathbf , \mathbf_ \rangle \mathbf_ = \mathbf^*\mathbf\mathbf. From this definition of \mathbf and linearity in the first argument of the inner product, : \langle \mathbf \mathbf , \mathbf \rangle = \sum_k \left, \langle \mathbf , \mathbf_k \rangle \ ^2 , which, when substituted in the frame condition inequality, yields : A \left\, \mathbf \right\, ^2 \leq \langle \mathbf \mathbf , \mathbf \rangle \leq B \left\, \mathbf \right\, ^2 , for each \mathbf \in V. The frame operator \mathbf is
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...
,
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
, and has positive upper and lower bounds. The inverse \mathbf^ of \mathbf exists and it, too, is self-adjoint, positive definite, and has positive upper and lower bounds. The dual frame is defined by mapping each element of the frame with \mathbf^: : \tilde_ = \mathbf^ \mathbf_ To see that this makes sense, let \mathbf be an element of V and let : \mathbf = \sum_ \langle \mathbf , \mathbf_ \rangle \tilde_. Thus : \mathbf = \sum_ \langle \mathbf , \mathbf_ \rangle ( \mathbf^ \mathbf_ ) = \mathbf^ \left ( \sum_ \langle \mathbf , \mathbf_ \rangle \mathbf_ \right ) = \mathbf^ \mathbf \mathbf = \mathbf, which proves that : \mathbf = \sum_ \langle \mathbf , \mathbf_ \rangle \tilde_. Alternatively, we can let : \mathbf = \sum_ \langle \mathbf , \tilde_ \rangle \mathbf_. By inserting the above definition of \tilde_ and applying the properties of \mathbf and its inverse, : \mathbf = \sum_ \langle \mathbf , \mathbf^ \mathbf_ \rangle \mathbf_ = \sum_ \langle \mathbf^ \mathbf , \mathbf_ \rangle \mathbf_ = \mathbf (\mathbf^ \mathbf) = \mathbf which shows that : \mathbf = \sum_ \langle \mathbf , \tilde_ \rangle \mathbf_. The numbers \langle \mathbf , \tilde_ \rangle are called ''frame coefficients''. This derivation of a dual frame is a summary of Section 3 in the article by Duffin and Schaeffer. They use the term ''conjugate frame'' for what here is called a dual frame. The dual frame \ is called the ''canonical dual'' of \ because it acts similarly as a
dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with th ...
to a basis. When the frame \ is overcomplete, a vector \mathbf can be written as a linear combination of \ in more than one way. That is, there are different choices of coefficients \ such that \mathbf = \sum_ c_ \mathbf_. This allows us some freedom for the choice of coefficients \ other than \langle \mathbf , \tilde_ \rangle. It is necessary that the frame \ is overcomplete for other such coefficients \ to exist. If so, then there exist frames \ \neq \ for which : \mathbf = \sum_ \langle \mathbf , \mathbf_ \rangle \mathbf_ for all \mathbf \in V. We call \ a dual frame of \. Canonical duality is a reciprocity relation, i.e. if the frame \ is the canonical dual frame of \, then \ is the canonical dual frame of \.


See also

* ''k''-frame *
Biorthogonal wavelet A Biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogon ...
*
Orthogonal wavelet An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened one may end up with biorthogonal wavelets. Basics ...
*
Restricted isometry property In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence TaoE. J. Candes and T. Tao, "Decodin ...
*
Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
*
Harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
*
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
*
Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...


Notes


References

* * * * * * {{DEFAULTSORT:Frame Of A Vector Space Linear algebra Differential geometry Signal processing