Calculus on Euclidean space

In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.

Basic notions

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Functions in one real variable

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This section is a brief review of function theory in one-variable calculus.

A real-valued function   is continuous at   if it is approximately constant near  ; i.e.,

 

In contrast, the function   is differentiable at   if it is approximately linear near  ; i.e., there is some real number   such that

 [1]

(For simplicity, suppose  . Then the above means that   where   goes to 0 faster than h going to 0 and, in that sense,   behaves like  .)

The number   depends on   and thus is denoted as  . If   is differentiable on an open interval   and if   is a continuous function on  , then   is called a C1 function. More generally,   is called a Ck function if its derivative   is Ck-1 function. Taylor's theorem states that a Ck function is precisely a function that can be approximated by a polynomial of degree k.

If   is a C1 function and   for some  , then either   or  ; i.e., either   is strictly increasing or strictly decreasing in some open interval containing a. In particular,   is bijective for some open interval   containing  . The inverse function theorem then says that the inverse function   is differentiable on U with the derivatives: for  

 

Derivative of a map and chain rule

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For functions   defined in the plane or more generally on an Euclidean space  , it is necessary to consider functions that are vector-valued or matrix-valued. It is also conceptually helpful to do this in an invariant manner (i.e., a coordinate-free way). Derivatives of such maps at a point are then vectors or linear maps, not real numbers.

Let   be a map from an open subset   of   to an open subset   of  . Then the map   is said to be differentiable at a point   in   if there exists a (necessarily unique) linear transformation  , called the derivative of   at  , such that

 

where   is the application of the linear transformation   to  .[2] If   is differentiable at  , then it is continuous at   since

  as  .

As in the one-variable case, there is

Chain rule — [3] Let   be as above and   a map for some open subset   of  . If   is differentiable at   and   differentiable at  , then the composition   is differentiable at   with the derivative

 

This is proved exactly as for functions in one variable. Indeed, with the notation  , we have:

 

Here, since   is differentiable at  , the second term on the right goes to zero as  . As for the first term, it can be written as:

 

Now, by the argument showing the continuity of   at  , we see   is bounded. Also,   as   since   is continuous at  . Hence, the first term also goes to zero as   by the differentiability of   at  .  

The map   as above is called continuously differentiable or   if it is differentiable on the domain and also the derivatives vary continuously; i.e.,   is continuous.

Corollary — If   are continuously differentiable, then   is continuously differentiable.

As a linear transformation,   is represented by an  -matrix, called the Jacobian matrix   of   at   and we write it as:

 

Taking   to be  ,   a real number and   the j-th standard basis element, we see that the differentiability of   at   implies:

 

where   denotes the i-th component of  . That is, each component of   is differentiable at   in each variable with the derivative  . In terms of Jacobian matrices, the chain rule says  ; i.e., as  ,

 

which is the form of the chain rule that is often stated.

A partial converse to the above holds. Namely, if the partial derivatives   are all defined and continuous, then   is continuously differentiable.[4] This is a consequence of the mean value inequality:

Mean value inequality — [5] Given the map   as above and points   in   such that the line segment between   lies in  , if   is continuous on   and is differentiable on the interior, then, for any vector  ,

 

where  

(This version of mean value inequality follows from mean value inequality in Mean value theorem § Mean value theorem for vector-valued functions applied to the function  , where the proof on mean value inequality is given.)

Indeed, let  . We note that, if  , then

 

For simplicity, assume   (the argument for the general case is similar). Then, by mean value inequality, with the operator norm  ,

 

which implies   as required.  

Example: Let   be the set of all invertible real square matrices of size n. Note   can be identified as an open subset of   with coordinates  . Consider the function   = the inverse matrix of   defined on  . To guess its derivatives, assume   is differentiable and consider the curve   where   means the matrix exponential of  . By the chain rule applied to  , we have:

 .

Taking  , we get:

 .

Now, we then have:[6]

 

Since the operator norm is equivalent to the Euclidean norm on   (any norms are equivalent to each other), this implies   is differentiable. Finally, from the formula for  , we see the partial derivatives of   are smooth (infinitely differentiable); whence,   is smooth too.

Higher derivatives and Taylor formula

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If   is differentiable where   is an open subset, then the derivatives determine the map  , where   stands for homomorphisms between vector spaces; i.e., linear maps. If   is differentiable, then  . Here, the codomain of   can be identified with the space of bilinear maps by:

 

where   and   is bijective with the inverse   given by  .[a] In general,   is a map from   to the space of  -multilinear maps  .

Just as   is represented by a matrix (Jacobian matrix), when   (a bilinear map is a bilinear form), the bilinear form   is represented by a matrix called the Hessian matrix of   at  ; namely, the square matrix   of size   such that  , where the paring refers to an inner product of  , and   is none other than the Jacobian matrix of  . The  -th entry of   is thus given explicitly as  .

Moreover, if   exists and is continuous, then the matrix   is symmetric, the fact known as the symmetry of second derivatives.[7] This is seen using the mean value inequality. For vectors   in  , using mean value inequality twice, we have:

 

which says

 

Since the right-hand side is symmetric in  , so is the left-hand side:  . By induction, if   is  , then the k-multilinear map   is symmetric; i.e., the order of taking partial derivatives does not matter.[7]

As in the case of one variable, the Taylor series expansion can then be proved by integration by parts:

 

Taylor's formula has an effect of dividing a function by variables, which can be illustrated by the next typical theoretical use of the formula.

Example:[8] Let   be a linear map between the vector space   of smooth functions on   with rapidly decreasing derivatives; i.e.,   for any multi-index  . (The space   is called a Schwartz space.) For each   in  , Taylor's formula implies we can write:

 

with  , where   is a smooth function with compact support and  . Now, assume   commutes with coordinates; i.e.,  . Then

 .

Evaluating the above at  , we get   In other words,   is a multiplication by some function  ; i.e.,  . Now, assume further that   commutes with partial differentiations. We then easily see that   is a constant;   is a multiplication by a constant.

(Aside: the above discussion almost proves the Fourier inversion formula. Indeed, let   be the Fourier transform and the reflection; i.e.,  . Then, dealing directly with the integral that is involved, one can see   commutes with coordinates and partial differentiations; hence,   is a multiplication by a constant. This is almost a proof since one still has to compute this constant.)

A partial converse to the Taylor formula also holds; see Borel's lemma and Whitney extension theorem.

Inverse function theorem and submersion theorem

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Inverse function theorem — Let   be a map between open subsets   in  . If   is continuously differentiable (or more generally  ) and   is bijective, there exists neighborhoods   of   and the inverse   that is continuously differentiable (or respectively  ).

A  -map with the  -inverse is called a  -diffeomorphism. Thus, the theorem says that, for a map   satisfying the hypothesis at a point  ,   is a diffeomorphism near   For a proof, see Inverse function theorem § A proof using successive approximation.

The implicit function theorem says:[9] given a map  , if  ,   is   in a neighborhood of   and the derivative of   at   is invertible, then there exists a differentiable map   for some neighborhoods   of   such that  . The theorem follows from the inverse function theorem; see Inverse function theorem § Implicit function theorem.

Another consequence is the submersion theorem.

Integrable functions on Euclidean spaces

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A partition of an interval   is a finite sequence  . A partition   of a rectangle   (product of intervals) in   then consists of partitions of the sides of  ; i.e., if  , then   consists of   such that   is a partition of  .[10]

Given a function   on  , we then define the upper Riemann sum of it as:

 

where

  •   is a partition element of  ; i.e.,   when   is a partition of  .[11]
  • The volume   of   is the usual Euclidean volume; i.e.,  .

The lower Riemann sum   of   is then defined by replacing   by  . Finally, the function   is called integrable if it is bounded and  . In that case, the common value is denoted as  .[12]

A subset of   is said to have measure zero if for each  , there are some possibly infinitely many rectangles   whose union contains the set and  [13]

A key theorem is

Theorem — [14] A bounded function   on a closed rectangle is integrable if and only if the set   has measure zero.

The next theorem allows us to compute the integral of a function as the iteration of the integrals of the function in one-variables:

Fubini's theorem — If   is a continuous function on a closed rectangle   (in fact, this assumption is too strong), then

 

In particular, the order of integrations can be changed.

Finally, if   is a bounded open subset and   a function on  , then we define   where   is a closed rectangle containing   and   is the characteristic function on  ; i.e.,   if   and   if   provided   is integrable.[15]

Surface integral

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If a bounded surface   in   is parametrized by   with domain  , then the surface integral of a measurable function   on   is defined and denoted as:

 

If   is vector-valued, then we define

 

where   is an outward unit normal vector to  . Since  , we have:

 

Vector analysis

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Tangent vectors and vector fields

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Let   be a differentiable curve. Then the tangent vector to the curve   at   is a vector   at the point   whose components are given as:

 .[16]

For example, if   is a helix, then the tangent vector at t is:

 

It corresponds to the intuition that the a point on the helix moves up in a constant speed.

If   is a differentiable curve or surface, then the tangent space to   at a point p is the set of all tangent vectors to the differentiable curves   with  .

A vector field X is an assignment to each point p in M a tangent vector   to M at p such that the assignment varies smoothly.

Differential forms

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The dual notion of a vector field is a differential form. Given an open subset   in  , by definition, a differential 1-form (often just 1-form)   is an assignment to a point   in   a linear functional   on the tangent space   to   at   such that the assignment varies smoothly. For a (real or complex-valued) smooth function  , define the 1-form   by: for a tangent vector   at  ,

 

where   denotes the directional derivative of   in the direction   at  .[17] For example, if   is the  -th coordinate function, then  ; i.e.,   are the dual basis to the standard basis on  . Then every differential 1-form   can be written uniquely as

 

for some smooth functions   on   (since, for every point  , the linear functional   is a unique linear combination of   over real numbers). More generally, a differential k-form is an assignment to a point   in   a vector   in the  -th exterior power   of the dual space   of   such that the assignment varies smoothly.[17] In particular, a 0-form is the same as a smooth function. Also, any  -form   can be written uniquely as:

 

for some smooth functions  .[17]

Like a smooth function, we can differentiate and integrate differential forms. If   is a smooth function, then   can be written as:[18]

 

since, for  , we have:  . Note that, in the above expression, the left-hand side (whence the right-hand side) is independent of coordinates  ; this property is called the invariance of differential.

The operation   is called the exterior derivative and it extends to any differential forms inductively by the requirement (Leibniz rule)

 

where   are a p-form and a q-form.

The exterior derivative has the important property that  ; that is, the exterior derivative   of a differential form   is zero. This property is a consequence of the symmetry of second derivatives (mixed partials are equal).

Boundary and orientation

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A circle can be oriented clockwise or counterclockwise. Mathematically, we say that a subset   of   is oriented if there is a consistent choice of normal vectors to   that varies continuously. For example, a circle or, more generally, an n-sphere can be oriented; i.e., orientable. On the other hand, a Möbius strip (a surface obtained by identified by two opposite sides of the rectangle in a twisted way) cannot oriented: if we start with a normal vector and travel around the strip, the normal vector at end will point to the opposite direction.

Proposition — A bounded differentiable region   in   of dimension   is oriented if and only if there exists a nowhere-vanishing  -form on   (called a volume form).

The proposition is useful because it allows us to give an orientation by giving a volume form.

Integration of differential forms

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If   is a differential n-form on an open subset M in   (any n-form is that form), then the integration of it over   with the standard orientation is defined as:

 

If M is given the orientation opposite to the standard one, then   is defined as the negative of the right-hand side.

Then we have the fundamental formula relating exterior derivative and integration:

Stokes' formula — For a bounded region   in   of dimension   whose boundary is a union of finitely many  -subsets, if   is oriented, then

 

for any differential  -form   on the boundary   of  .

Here is a sketch of proof of the formula.[19] If   is a smooth function on   with compact support, then we have:

 

(since, by the fundamental theorem of calculus, the above can be evaluated on boundaries of the set containing the support.) On the other hand,

 

Let   approach the characteristic function on  . Then the second term on the right goes to   while the first goes to  , by the argument similar to proving the fundamental theorem of calculus.  

The formula generalizes the fundamental theorem of calculus as well as Stokes' theorem in multivariable calculus. Indeed, if   is an interval and  , then   and the formula says:

 .

Similarly, if   is an oriented bounded surface in   and  , then   and similarly for   and  . Collecting the terms, we thus get:

 

Then, from the definition of the integration of  , we have   where   is the vector-valued function and  . Hence, Stokes’ formula becomes

 

which is the usual form of the Stokes' theorem on surfaces. Green’s theorem is also a special case of Stokes’ formula.

Stokes' formula also yields a general version of Cauchy's integral formula. To state and prove it, for the complex variable   and the conjugate  , let us introduce the operators

 

In these notations, a function   is holomorphic (complex-analytic) if and only if   (the Cauchy–Riemann equations). Also, we have:

 

Let   be a punctured disk with center  . Since   is holomorphic on  , We have:

 .

By Stokes’ formula,

 

Letting   we then get:[20][21]

 

Winding numbers and Poincaré lemma

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A differential form   is called closed if   and is called exact if   for some differential form   (often called a potential). Since  , an exact form is closed. But the converse does not hold in general; there might be a non-exact closed form. A classic example of such a form is:[22]

 ,

which is a differential form on  . Suppose we switch to polar coordinates:   where  . Then

 

This does not show that   is exact: the trouble is that   is not a well-defined continuous function on  . Since any function   on   with   differ from   by constant, this means that   is not exact. The calculation, however, shows that   is exact, for example, on   since we can take   there.

There is a result (Poincaré lemma) that gives a condition that guarantees closed forms are exact. To state it, we need some notions from topology. Given two continuous maps   between subsets of   (or more generally topological spaces), a homotopy from   to   is a continuous function   such that   and  . Intuitively, a homotopy is a continuous variation of one function to another. A loop in a set   is a curve whose starting point coincides with the end point; i.e.,   such that  . Then a subset of   is called simply connected if every loop is homotopic to a constant function. A typical example of a simply connected set is a disk  . Indeed, given a loop  , we have the homotopy   from   to the constant function  . A punctured disk, on the other hand, is not simply connected.

Poincaré lemma — If   is a simply connected open subset of  , then each closed 1-form on   is exact.

Geometry of curves and surfaces

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Moving frame

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Vector fields   on   are called a frame field if they are orthogonal to each other at each point; i.e.,   at each point.[23] The basic example is the standard frame  ; i.e.,   is a standard basis for each point   in  . Another example is the cylindrical frame

 [24]

For the study of the geometry of a curve, the important frame to use is a Frenet frame   on a unit-speed curve   given as:

The Gauss–Bonnet theorem

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The Gauss–Bonnet theorem relates the topology of a surface and its geometry.

The Gauss–Bonnet theorem — [25] For each bounded surface   in  , we have:

 

where   is the Euler characteristic of   and   the curvature.

Calculus of variations

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Method of Lagrange multiplier

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Lagrange multiplier — [26] Let   be a differentiable function from an open subset of   such that   has rank   at every point in  . For a differentiable function  , if   attains either a maximum or minimum at a point   in  , then there exists real numbers   such that

 .

In other words,   is a stationary point of  .

The set   is usually called a constraint.

Example:[27] Suppose we want to find the minimum distance between the circle   and the line  . That means that we want to minimize the function  , the square distance between a point   on the circle and a point   on the line, under the constraint  . We have:

 
 

Since the Jacobian matrix of   has rank 2 everywhere on  , the Lagrange multiplier gives:

 

If  , then  , not possible. Thus,   and

 

From this, it easily follows that   and  . Hence, the minimum distance is   (as a minimum distance clearly exists).

Here is an application to linear algebra.[28] Let   be a finite-dimensional real vector space and   a self-adjoint operator. We shall show   has a basis consisting of eigenvectors of   (i.e.,   is diagonalizable) by induction on the dimension of  . Choosing a basis on   we can identify   and   is represented by the matrix  . Consider the function  , where the bracket means the inner product. Then  . On the other hand, for  , since   is compact,   attains a maximum or minimum at a point   in  . Since  , by Lagrange multiplier, we find a real number   such that   But that means  . By inductive hypothesis, the self-adjoint operator  ,   the orthogonal complement to  , has a basis consisting of eigenvectors. Hence, we are done.  .

Weak derivatives

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Up to measure-zero sets, two functions can be determined to be equal or not by means of integration against other functions (called test functions). Namely, the following sometimes called the fundamental lemma of calculus of variations:

Lemma[29] — If   are locally integrable functions on an open subset   such that

 

for every   (called a test function). Then   almost everywhere. If, in addition,   are continuous, then  .

Given a continuous function  , by the lemma, a continuously differentiable function   is such that   if and only if

 

for every  . But, by integration by parts, the partial derivative on the left-hand side of   can be moved to that of  ; i.e.,

 

where there is no boundary term since   has compact support. Now the key point is that this expression makes sense even if   is not necessarily differentiable and thus can be used to give sense to a derivative of such a function.

Note each locally integrable function   defines the linear functional   on   and, moreover, each locally integrable function can be identified with such linear functional, because of the early lemma. Hence, quite generally, if   is a linear functional on  , then we define   to be the linear functional   where the bracket means  . It is then called the weak derivative of   with respect to  . If   is continuously differentiable, then the weak derivate of it coincides with the usual one; i.e., the linear functional   is the same as the linear functional determined by the usual partial derivative of   with respect to  . A usual derivative is often then called a classical derivative. When a linear functional on   is continuous with respect to a certain topology on  , such a linear functional is called a distribution, an example of a generalized function.

A classic example of a weak derivative is that of the Heaviside function  , the characteristic function on the interval  .[30] For every test function  , we have:

 

Let   denote the linear functional  , called the Dirac delta function (although not exactly a function). Then the above can be written as:

 

Cauchy's integral formula has a similar interpretation in terms of weak derivatives. For the complex variable  , let  . For a test function  , if the disk   contains the support of  , by Cauchy's integral formula, we have:

 

Since  , this means:

 

or

 [31]

In general, a generalized function is called a fundamental solution for a linear partial differential operator if the application of the operator to it is the Dirac delta. Hence, the above says   is the fundamental solution for the differential operator  .

Hamilton–Jacobi theory

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Calculus on manifolds

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Definition of a manifold

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This section requires some background in general topology.

A manifold is a Hausdorff topological space that is locally modeled by an Euclidean space. By definition, an atlas of a topological space   is a set of maps  , called charts, such that

  •   are an open cover of  ; i.e., each   is open and  ,
  •   is a homeomorphism and
  •   is smooth; thus a diffeomorphism.

By definition, a manifold is a second-countable Hausdorff topological space with a maximal atlas (called a differentiable structure); "maximal" means that it is not contained in strictly larger atlas. The dimension of the manifold   is the dimension of the model Euclidean space  ; namely,   and a manifold is called an n-manifold when it has dimension n. A function on a manifold   is said to be smooth if   is smooth on   for each chart   in the differentiable structure.

A manifold is paracompact; this has an implication that it admits a partition of unity subordinate to a given open cover.

If   is replaced by an upper half-space  , then we get the notion of a manifold-with-boundary. The set of points that map to the boundary of   under charts is denoted by   and is called the boundary of  . This boundary may not be the topological boundary of  . Since the interior of   is diffeomorphic to  , a manifold is a manifold-with-boundary with empty boundary.

The next theorem furnishes many examples of manifolds.

Theorem — [32] Let   be a differentiable map from an open subset   such that   has rank   for every point   in  . Then the zero set   is an  -manifold.

For example, for  , the derivative   has rank one at every point   in  . Hence, the n-sphere   is an n-manifold.

The theorem is proved as a corollary of the inverse function theorem.

Many familiar manifolds are subsets of  . The next theoretically important result says that there is no other kind of manifolds. An immersion is a smooth map whose differential is injective. An embedding is an immersion that is homeomorphic (thus diffeomorphic) to the image.

Whitney's embedding theorem — Each  -manifold can be embedded into  .

The proof that a manifold can be embedded into   for some N is considerably easier and can be readily given here. It is known [citation needed] that a manifold has a finite atlas  . Let   be smooth functions such that   and   cover   (e.g., a partition of unity). Consider the map

 

It is easy to see that   is an injective immersion. It may not be an embedding. To fix that, we shall use:

 

where   is a smooth proper map. The existence of a smooth proper map is a consequence of a partition of unity. See [1] for the rest of the proof in the case of an immersion.  

Nash's embedding theorem says that, if   is equipped with a Riemannian metric, then the embedding can be taken to be isometric with an expense of increasing  ; for this, see this T. Tao's blog.

Tubular neighborhood and transversality

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A technically important result is:

Tubular neighborhood theorem — Let M be a manifold and   a compact closed submanifold. Then there exists a neighborhood   of   such that   is diffeomorphic to the normal bundle   to   and   corresponds to the zero section of   under the diffeomorphism.

This can be proved by putting a Riemannian metric on the manifold  . Indeed, the choice of metric makes the normal bundle   a complementary bundle to  ; i.e.,   is the direct sum of   and  . Then, using the metric, we have the exponential map   for some neighborhood   of   in the normal bundle   to some neighborhood   of   in  . The exponential map here may not be injective but it is possible to make it injective (thus diffeomorphic) by shrinking   (for now, see [2]).


Integration on manifolds and distribution densities

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The starting point for the topic of integration on manifolds is that there is no invariant way to integrate functions on manifolds. This may be obvious if we asked: what is an integration of functions on a finite-dimensional real vector space? (In contrast, there is an invariant way to do differentiation since, by definition, a manifold comes with a differentiable structure). There are several ways to introduce integration theory to manifolds:

  • Integrate differential forms.
  • Do integration against some measure.
  • Equip a manifold with a Riemannian metric and do integration against such a metric.

For example, if a manifold is embedded into an Euclidean space  , then it acquires the Lebesgue measure restricting from the ambient Euclidean space and then the second approach works. The first approach is fine in many situations but it requires the manifold to be oriented (and there is a non-orientable manifold that is not pathological). The third approach generalizes and that gives rise to the notion of a density.

Generalizations

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Extensions to infinite-dimensional normed spaces

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The notions like differentiability extend to normed spaces.

See also

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Notes

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  1. ^ This is just the tensor-hom adjunction.

Citations

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  1. ^ Spivak 1965, Ch 2. Basic definitions.
  2. ^ Hörmander 2015, Definition 1.1.4.
  3. ^ Hörmander 2015, (1.1.3.)
  4. ^ Hörmander 2015, Theorem 1.1.6.
  5. ^ Hörmander 2015, (1.1.2)'
  6. ^ Hörmander 2015, p. 8
  7. ^ a b Hörmander 2015, Theorem 1.1.8.
  8. ^ Hörmander 2015, Lemma 7.1.4.
  9. ^ Spivak 1965, Theorem 2-12.
  10. ^ Spivak 1965, p. 46
  11. ^ Spivak 1965, p. 47
  12. ^ Spivak 1965, p. 48
  13. ^ Spivak 1965, p. 50
  14. ^ Spivak 1965, Theorem 3-8.
  15. ^ Spivak 1965, p. 55
  16. ^ Spivak 1965, Exercise 4.14.
  17. ^ a b c Spivak 1965, p. 89
  18. ^ Spivak 1965, Theorem 4-7.
  19. ^ Hörmander 2015, p. 151
  20. ^ Theorem 1.2.1. in Hörmander, Lars (1990). An Introduction to Complex Analysis in Several Variables (Third ed.). North Holland..
  21. ^ Spivak 1965, Exercise 4-33.
  22. ^ Spivak 1965, p. 93
  23. ^ O'Neill 2006, Definition 6.1.
  24. ^ O'Neill 2006, Example 6.2. (1)
  25. ^ O'Neill 2006, Theorem 6.10.
  26. ^ Spivak 1965, Exercise 5-16.
  27. ^ Edwards 1994, Ch. II, $ 5. Example 9.
  28. ^ Spivak 1965, Exercise 5-17.
  29. ^ Hörmander 2015, Theorem 1.2.5.
  30. ^ Hörmander 2015, Example 3.1.2.
  31. ^ Hörmander 2015, p. 63
  32. ^ Spivak 1965, Theorem 5-1.

References

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