Inner product space

In linear algebra and 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 vector space with inner product, sporadically inner product space ) refers. A distinction is made between Euclidean (vector ) spaces in 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. But the nomenclature is not uniform. Some authors include the unitary vector space to the real case (which can be so construed as limiting ) with one, and sometimes it's the other way around, that is the complex vector spaces are called Euclidean.

The importance of Prähilberträume is that the dot product in analogy to the analytical geometry allows the adoption of the terms length ( on the induced norm) and angle. Each pre-Hilbert space is therefore a normed vector space. By the length ( norm) and a distance ( metric ) is defined. Is the space with respect to this metric completely, it is a Hilbert space. Hilbert spaces are the most direct generalization of Euclidean geometry on infinite-dimensional spaces.

• 3.1 Real and complex numbers
• 3.2 vectors of finite dimension
• 3.3 Continuous Functions
• 3.4 Hilbert space

Formal definition

A significant aspect of the classical ( Euclidean ) geometry is the possibility of measuring lengths and angles. In the axiomatic justification of the geometry, this is ensured by the axioms of congruence. Performs to a Cartesian coordinate system, the lengths and angles with the help of the inner product can be calculated from the coordinates. In order to transfer lengths and angles from the Euclidean space to general vector spaces, if you drop the reference to a particular base and characterized by abstract inner products are crucial for the length measurement properties. This leads to the following definition:

Scalar product

Be a vector space over the field of real or complex numbers. A scalar or inner product is a positive definite hermitian sesquilinear, ie a mapping

For all, and for all fulfilled the following axiomatic conditions:

• (1 ) ( non-negative );
• (2 ) ( definite);
• (3 ) ( Hermitian );
• (4a) and (4b ) (linear in the second argument ).

Of the conditions (3) and (4) follows

• (5a) and (5b ) ( semilinear in the first argument )

For (4) and ( 5) is a sesquilinear.

Remarks:

• The overline in the third axiom means complex conjugation. In a real vector space (ie, when ), the complex conjugate of no effect. It follows:

• (3 ') (balanced)

• This definition, according to which the scalar product is semi- linear in the first argument and linear in the second, prevails in theoretical physics. However, often the condition ( 4 ) is selected in the first place for the second argument:

• ( 4a ' ) (linearity in the first argument ) and therefore
• ( 5a ' ) (Semi -linearity in the second argument )

Pre-Hilbert space

A pre-Hilbert space is then a real or complex vector space with a scalar product.

Notation

The inner product is sometimes written with a dot as a multiplication sign. In the French literature a subscript point is in use. In the functional analysis, or whenever else the connection of the inner product with linear functions ( and in particular the duality between and ) should be emphasized, it is preferable to notation. Derived from this is the Bra- Ket notation, which is often used in quantum mechanics.

As with the normal multiplication the multiplication sign can also be completely omitted if no misunderstandings are to be feared; this is particularly the case in texts in which vectors by vector arrows in bold or by underlining are identified and therefore can not be confused with scalars:

Examples

Real and complex numbers

The vector space of real numbers with the scalar product and the vector space of complex numbers with the scalar product are simple examples of Prähilberträume.

Vectors of finite dimension

For is

The standard scalar product defined, which makes the vector space not only to a pre-Hilbert space, but also to a Hilbert space, then there exists completeness.

Continuous Functions

Another example of a real pre-Hilbert space is the space of all continuous functions of a real interval according to the inner product

With a continuous positive weight function (or "Assignment " ) is (instead it is sufficient to call with weak additional conditions ). An orthogonal basis of this space is called orthogonal system; Examples of such systems are the trigonometric function used in Fourier series, the Legendre polynomials, Chebyshev polynomials, the polynomials Laguerre that Hermite polynomials, etc.

Hilbert space

Every Hilbert space is a pre-Hilbert space.

Induced norm

Each inner product induced on the underlying vector space of a standard

The proof of the triangle inequality for the mapping defined in this way requires to be non -trivial intermediate step, the Cauchy- Schwarz inequality

With the induced norm of any pre-Hilbert space is a normed space, in which the parallelogram law

Applies. Conversely, with the set of Jordan -von Neumann that every normed space in which the parallelogram law is met, a pre-Hilbert space is. The associated scalar product can be defined by a polarization formula, in the real case, for example, about

Position in the hierarchy of mathematical structures

With the induced by the inner product of each standard inner product space is a normed space, so that a metric space, so that a topological space; ie, it has both a geometric and a topological structure.

A complete inner product space is called a Hilbert space. Each pre-Hilbert space can be completed in a unique way (up to isometric isomorphism ) to a Hilbert space.

Generalizations: metric tensor, Bilinearräume, theory of relativity

From the standpoint of the tensor algebra of the inner product can

With the notation as a second-rank tensor

Be construed, where the tensor product and the dual space of designated; is called metric tensor or short metric. The requirement that the inner product must be positive definite, this means that in any coordinate system associated to the matrix is positive definite, so only has positive eigenvalues ​​.

A generalized inner product of spaces are Bilinearräume in which the inner product is replaced by a Hermitian, or bilinear form, which is not necessary for positive definite. An important example is the Minkowski space of special relativity, the metric in the common convention has eigenvalues ​​with the sign.