Prehilbert dream

In the linear algebra and in the functional analysis is a real or complex vector space on which an inner product ( dot product ) is defined as a pre-Hilbert space (also prähilbertscher space ) or scalar product (also a vector space with the inner product isolated, and inner product space ), respectively. A distinction is made between Euclidean (vector) spaces in the real and unitary (vector) spaces in the complex case. The finite-dimensional ( n -dimensional) Euclidean vector spaces are models for the n -dimensional Euclidean space . However, the nomenclature is not uniform. Some authors include the real case in the unitary vector space (which can be regarded as a restriction), and sometimes it is also the other way round, that is, the complex vector spaces are also called Euclidean.

The meaning of the Prähilbert spaces lies in the fact that the scalar product allows the introduction of the terms length (via the induced norm ) and angle in analogy to analytical geometry . Every Prähilbert space is therefore a normalized vector space . The length (norm) also defines a distance ( metric ). If the room is with respect to this metric completely , it is a Hilbert space . Hilbert spaces are the direct generalization of Euclidean geometry to infinite dimensional spaces.

Formal definition

An essential aspect of classical (Euclidean) geometry is the ability to measure lengths and angles. In the axiomatic foundation of geometry this is secured by the axioms of congruence. If a Cartesian coordinate system is introduced, the lengths and angles can be calculated from the coordinates with the aid of the scalar product . In order to transfer lengths and angles from Euclidean space to general vector spaces, the reference to a certain basis is dropped and abstract inner products are characterized by the properties that are decisive for length measurement. This leads to the following definition:

Scalar product

→ Main article : scalar product

Let be a vector space over the field of real or complex numbers. A scalar product or inner product is a positively definite Hermitian sesquilinear form , that is, a mapping ${\displaystyle V}$ ${\displaystyle \mathbb {K} }$

${\displaystyle \langle \cdot ,\cdot \rangle \colon V\times V\to {\mathbb {K} }}$,

common to all , , from and for all the following axiomatic conditions met: ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle V}$ ${\displaystyle \lambda \in {\mathbb {K} }}$

• (1)      (not negative); ${\displaystyle \langle {x},{x}\rangle \geq 0}$
• (2)      ( definite ); ${\displaystyle \langle {x},{x}\rangle =0\Leftrightarrow {x}={0}}$
• (3)      (Hermitian); ${\displaystyle \langle {x},{y}\rangle ={\overline {\langle {y},{x}\rangle }}}$
• (4a)    and (4b)    ( linear in the second argument). ${\displaystyle \langle {x},\lambda {y}\rangle =\lambda \langle {x},{y}\rangle }$
  ${\displaystyle \langle {x},{y}+{z}\rangle =\langle {x},{y}\rangle +\langle {x},{z}\rangle }$

From the conditions (3) and (4) it follows

• (5a)    and (5b)    ( semilinear in the first argument) ${\displaystyle \langle \lambda {x},{y}\rangle ={\overline {\lambda }}\langle {x},{y}\rangle }$
  ${\displaystyle \langle {x}+{z},{y}\rangle =\langle {x},{y}\rangle +\langle {z},{y}\rangle }$

Because of (4) and (5) is a sesquilinear form . ${\displaystyle \langle \cdot ,\cdot \rangle }$

Remarks:

• The overline in the third axiom means complex conjugation . In a real vector space (i.e. if is) the complex conjugation has no effect. It follows: ${\displaystyle {\mathbb {K} }=\mathbb {R} }$

In a real vector space, (3) is equivalent to

• (3 ')    (symmetrical) ${\displaystyle \langle {x},{y}\rangle =\langle {y},{x}\rangle }$

and the scalar product is a symmetric bilinear form .

• This definition, according to which the scalar product is semilinear in the first argument and linear in the second, prevails in theoretical physics. Often, however, condition (4a) is chosen for the first instead of the second argument:

• (4a ')    (linearity in the first argument) and therefore ${\displaystyle \langle \lambda {x},{y}\rangle =\lambda \langle {x},{y}\rangle }$
• (5a ')    (semi-linearity in the second argument) ${\displaystyle \langle {x},\lambda {y}\rangle ={\overline {\lambda }}\langle {x},{y}\rangle }$

So you have to be careful whether the inner product in a given text is linear in the first or in the second argument.

Prehilbert dream

A Prähilbert space is then a real or complex vector space together with a scalar product.

notation

The inner product is sometimes written with a point as the multiplication sign: . In French literature a subscript point is common: . In functional analysis , or whenever the connection between the inner product and linear functions (and in particular the duality between and ) needs to be emphasized, the notation is preferred . Is derived from the bra-ket notation in the, quantum mechanics is often used: . ${\displaystyle x\cdot y}$ ${\displaystyle x.y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \langle x,y\rangle }$ ${\displaystyle \langle x\mid y\rangle }$

As with normal multiplication, the multiplication sign can be omitted entirely if there is no risk of misunderstanding; this is especially the case in texts in which vectors are indicated by vector arrows, by bold type or by underlining and therefore cannot be confused with scalars:

${\displaystyle \mathbf {x} \cdot \mathbf {y} =\mathbf {xy} }$ is an inner product

${\displaystyle a\mathbf {x} }$on the other hand is the multiplication of the vector by the scalar . ${\displaystyle \mathbf {x} }$ ${\displaystyle a}$

Examples

Real and complex numbers

The vector space of the real numbers with the scalar product and the vector space of the complex numbers with the scalar product are simple examples of prehilbert spaces. ${\displaystyle \mathbb {R} }$ ${\displaystyle \langle x,y\rangle =xy}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \langle x,y\rangle ={\bar {x}}{y}}$

Vectors of finite dimension

For is through ${\displaystyle x,y\in {\mathbb {K} }^{n}}$

${\displaystyle \langle x,y\rangle =\sum _{j=1}^{n}{\bar {x}}_{j}{y_{j}}}$

defines the standard scalar product, which makes the vector space not only a Prehilbert space, but also a Hilbert space, since then completeness is present. ${\displaystyle {\mathbb {K} }^{n}}$

Continuous functions

Another example of a real Prehilbert space is the space of all continuous functions from a real interval to with the inner product ${\displaystyle [a,b]}$ ${\displaystyle \mathbb {R} }$

${\displaystyle \langle f,g\rangle =\int _{a}^{b}p(x)f(x)g(x)\ {\rm {d}}x}$,

where a steady positive weight function (or "occupancy") is (instead of it is sufficient to demand with weak additional conditions). An orthogonal basis of this space is called an orthogonal system of functions ; Examples of such systems of functions are the trigonometric functions used in Fourier series , the Legendre polynomials , the Chebyshev polynomials , the Laguerre polynomials , the Hermite polynomials , etc. ${\displaystyle p}$ ${\displaystyle p(x)>0}$ ${\displaystyle p(x)\geq 0}$

Hilbert dream

Every Hilbert dream is a Prehilbert dream.

Induced norm

→ Main article : scalar product norm

Every inner product induces a norm on the underlying vector space

${\displaystyle \|x\|={\sqrt {\langle {x},{x}\rangle }}}$.

The proof of the triangle inequality for the mapping thus defined requires the Cauchy-Schwarz inequality as a nontrivial intermediate step

${\displaystyle |\langle x,y\rangle |\leq \|x\|\cdot \|y\|}$.

With the induced norm, every Prähilbert space is a normalized space in which the parallelogram equation

${\displaystyle 2\left(\|x\|^{2}+\|y\|^{2}\right)=\|x+y\|^{2}+\|x-y\|^{2}}$.

applies. Conversely, with Jordan-von Neumann's theorem , every normalized space in which the parallelogram equation is satisfied is a Prehilbert space. The associated scalar product can be defined by a polarization formula, in the real case for example via

${\displaystyle \langle x,y\rangle ={1 \over 4}\left({\|x+y\|}^{2}-{\|x-y\|}^{2}\right)}$.

Classification in the hierarchy of mathematical structures

Overview of abstract spaces in mathematics. An arrow is to be understood as an implication, i.e. a space at the beginning of the arrow is also a space at the end of the arrow.

With the norm induced by the inner product, every interior product space is a standardized space , thus also a metric space , and thus also a topological space ; so it has both a geometric and a topological structure.

A complete interior product space is called a Hilbert space . Each Prähilbert space can be completed in a unique way (except for isometric isomorphism ) to a Hilbert space.

Generalizations: metric tensor, bilinear spaces, relativity theory

From the standpoint of tensor algebra , the inner product can

${\displaystyle g\colon V\times V\to {\mathbb {K} }}$

with the notation as a second order tensor ${\displaystyle g({x},{x}):=\langle {x},{x}\rangle }$

${\displaystyle g\in V^{*}\otimes V^{*}}$

can be understood, where the tensor product and the dual space of denotes; is called the metric tensor or metric for short. The requirement that the inner product must be positive definite means that in any coordinate system the associated matrix is positive definite, i.e. has only positive eigenvalues. ${\displaystyle \otimes }$ ${\displaystyle V^{\ast }}$ ${\displaystyle V}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle g_{ik}}$

A generalization of interior product spaces are bilinear spaces in which the interior product is replaced by a Hermitian shape or bilinear shape that is not necessarily positive definite. An important example is the Minkowski space of the special theory of relativity, whose metric has eigenvalues ​​with the sign or . ${\displaystyle (-,+,+,+)}$ ${\displaystyle (+,-,-,-)}$

literature

• Dirk Werner : functional analysis , Springer-Verlag, Berlin, 2007, ISBN 978-3-540-72533-6

Individual evidence

1. ^ Günter Grosche, Viktor Ziegler, Eberhard Zeidler and Dorothea Ziegler: Teubner-Taschenbuch der Mathematik 2 . 8th edition. BG Teubner Verlag , 2003, ISBN 978-3-519-21008-5 , chapter 11.2, p. 354 ( online ).