Standardized space

In mathematics, a normalized space or normalized vector space is a vector space on which a norm is defined. Every normalized space is a metric space with the metric induced by the norm and a topological space with the topology induced by this metric . If a normalized space is complete , it is called a complete normalized space or Banach space . A normalized space can be derived from a Prehilbert space using the scalar product norm or from a vector space with a semi-norm as the factor space .

Standardized spaces are a central study object of functional analysis and play an important role in the solution structure of partial differential equations and integral equations .

definition

If there is a vector space over the field of real or complex numbers and a norm on , then the pair is called a normalized vector space . A norm is a mapping that assigns a non-negative real number to an element of the vector space and has the three properties of definiteness , absolute homogeneity and subadditivity . That is, is a norm if for all from- vector space and all from : ${\ displaystyle V}$ ${\ displaystyle \ mathbb {K}}$${\ displaystyle \ | \ cdot \ | \ colon V \ to \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle V}$${\ displaystyle (V, \ | \ cdot \ |)}$${\ displaystyle \ | \ cdot \ |}$${\ displaystyle \ | \ cdot \ | \ colon V \ to \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle x, y}$${\ displaystyle \ mathbb {K}}$${\ displaystyle \ lambda}$${\ displaystyle \ mathbb {K}}$

1. ${\ displaystyle \ | x \ | = 0 \ quad \ Leftrightarrow \ quad x = 0}$   (Definiteness)
2. ${\ displaystyle \ | \ lambda x \ | = | \ lambda | \ cdot \ | x \ |}$   (absolute homogeneity)
3. ${\ displaystyle \ | x + y \ | \ leq \ | x \ | + \ | y \ |}$   (Subadditivity, also called triangle inequality )

If it is clear which standard it is, you can do without its explicit specification and only write for the standardized space. ${\ displaystyle V}$

history

From 1896 onwards, Hermann Minkowski used finite-dimensional normed vector spaces to investigate number-theoretic questions according to today's terminology. The axiomatic definition of vector space did not gain acceptance until the 1920s. Minkowski found that in order to determine a distance that is compatible with the vector structure, it is only necessary to specify the calibration body. A calibration body is the set of all vectors with the norm or length less than or equal to one. For example, the solid sphere with radius one is a calibration body. Minkowski also found that the calibration body is a convex subset that is centrally symmetrical with respect to the origin of the coordinates , see Minkowski functional .

The standard symbol used today was first used by Erhard Schmidt in 1908. His work suggested that the expression should be understood as the distance between the vectors and . In a work published in 1918, Frigyes Riesz systematically used the norm symbol for the supremum norm in the space of continuous functions over a compact interval . ${\ displaystyle \ | \ cdot \ |}$${\ displaystyle \ | xy \ |}$ ${\ displaystyle x}$${\ displaystyle y}$

After preliminary work by Henri Lebesgue from 1910 and 1913, Stefan Banach developed the axiomatic definition of the norm or the standardized vector space in his dissertation from 1922. The complete normalized vector spaces, the Banach spaces , are named after him.

Examples

The following normalized spaces are all also complete:

• the space of real or complex numbers with the norm :${\ displaystyle ({\ mathbb {K}}, | \ cdot |)}$
• the space of real or complex vectors with the p -norm :${\ displaystyle ({\ mathbb {K}} ^ {n}, \ | \ cdot \ | _ {p})}$
• the space of real or complex matrices with the Frobenius norm :${\ displaystyle ({\ mathbb {K}} ^ {m, n}, \ | \ cdot \ | _ {F})}$
• the space of real- or complex-valued in p summable th potency Follow the p norm :${\ displaystyle (\ ell ^ {p}, \ | \ cdot \ | _ {\ ell ^ {p}})}$
• the space of real- or complex-valued bounded functions with the supremum norm :${\ displaystyle (B, \ | \ cdot \ | _ {\ sup})}$
• the space of real or complex valued continuous functions on a compact set of definitions with the maximum norm :${\ displaystyle (C ^ {0}, \ | \ cdot \ | _ {\ max})}$
• the space of real or complex-valued functions in the p- th power Lebesgue integrable with the L p -norm :${\ displaystyle (L ^ {p}, \ | \ cdot \ | _ {L ^ {p}})}$
• the space of real or complex valued functions limited m -fold continuously differentiable with the C m norm :${\ displaystyle (C ^ {m}, \ | \ cdot \ | _ {C ^ {m}})}$

The following example is complete if and only if the vector space is complete: ${\ displaystyle W}$

• the space of bounded linear operators between two real or complex vector spaces with the operator norm :${\ displaystyle ({\ mathfrak {L}} (V, W), \ | \ cdot \ | _ {{\ mathfrak {L}} (V, W)})}$

Related spaces

Classification of standardized spaces in different types of abstract spaces in mathematics

Special cases

Scalar product spaces

A norm can, but does not have to, be defined by a scalar product (inner product) . Each interior product space is with the norm induced by the scalar product${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$

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

a standardized space. A norm is induced by a scalar product if and only if the parallelogram equation is fulfilled in the resulting space .

A complete interior product space is called a Hilbert space .

Complete rooms

A normalized space is called complete if every Cauchy sequence in this space has a limit value. A completely normalized space is called a Banach space. Each room can be normalized by the formation of equivalence classes of Cauchy sequences complete . In this way a Banach space is obtained that contains the original space as a dense subspace .

Generalizations

Semi-normalized spaces

If there is only a semi-standard , one speaks of a semi- standardized space. From a space with a semi-norm, a normalized space is obtained as a factor space . For this purpose, elements and with each other are identified that fulfill In functional analysis, in addition to normalized spaces, one also considers vector spaces with a set of semi-norms and thus comes to the concept of locally convex space . ${\ displaystyle \ | \ cdot \ |}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ | xy \ | = 0}$

Metric and topological spaces

Any norm induced by

${\ displaystyle d (x, y) = \ | xy \ |}$

a metric . Every standardized space is therefore also a metric space and, with the standard topology, also a topological space . This defines topological terms such as limit value , Cauchy sequence , continuity and compactness in standardized spaces . So a sequence converges to a limit if and only if holds. The norm itself is a continuous mapping with respect to the topology induced by it. The metric space is a real generalization of the normalized space, as there are metric spaces in which (a) the metric cannot be represented by a norm and / or (b) is not a vector space at all, for example in the absence of an algebraic structure. ${\ displaystyle (V, d)}$ ${\ displaystyle {\ mathcal {T}}}$ ${\ displaystyle (V, {\ mathcal {T}})}$${\ displaystyle (x_ {n}) _ {n}}$${\ displaystyle x}$${\ displaystyle \ | x_ {n} -x \ | \ rightarrow 0}$${\ displaystyle (X, d)}$
${\ displaystyle d}$
${\ displaystyle X}$

Equivalent norms induce the same uniform structure and thus the same topology. In finite-dimensional vector spaces, all norms are equivalent to one another, but this is not the case in infinite-dimensional spaces.

A topological vector space is called normalizable if its topology can be generated from a norm. According to Kolmogoroff's criterion for normalization , the topology of a Hausdorff topological vector space is generated by a norm if its zero vector has a restricted and convex environment .

Spaces over valued bodies

The term normalized space can be understood in a more general way by allowing arbitrary vector spaces over evaluated bodies , i.e. bodies with an absolute value , instead of vector spaces over the body of real or complex numbers . ${\ displaystyle \ mathbb {K}}$${\ displaystyle (K, | \ cdot |)}$ ${\ displaystyle | \ cdot |}$