Inner product space questions

In summary, the conversation discusses the definition and properties of the inner product in relation to bilinear functions and mappings in vector spaces. It is clarified that the inner product is both conjugate linear and linear, making it a bilinear function. However, not all mappings from a vector space to its field can be considered inner products, especially in the case of infinite-dimensional vector spaces.
  • #1
quasar_4
290
0
Hello all,

I have two questions that are fairly general, but slightly hazy to me still. o:)

1) Can we consider the inner product to be a bilinear function, or not? I would like to think of it as a mapping from an ordered pair of vectors of some vector space V (i.e. VxV) to the field (F), and I know by definition the inner product is conjugate linear as a function of it's first entry (or second, depending on which text you use). But isn't the inner product also linear as a function of either entry whenever the other is held fixed, making it bilinear?

2) Can every mapping from a finite dimensional vector space V to it's field be considered an inner product of something? What about the infinite dimensional case?
 
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  • #2
1) Yes, obviously: what is a bilinear map?

2) No, equally obviously. If you allow mapping to mean linear functional then you need the Reitz representation theorem. But since very few vector spaces have inner products the answer is still 'NO', for equally obvious reasons.
 
  • #3


Hi there,

1) Yes, the inner product can be considered a bilinear function. As you mentioned, it is conjugate linear in one entry and linear in the other when the other is held fixed. This property makes it bilinear, as it satisfies the definition of a bilinear function.

2) No, not every mapping from a finite dimensional vector space V to its field can be considered an inner product. An inner product must satisfy certain properties, such as symmetry and positive definiteness. So not all mappings from V to its field will satisfy these properties and thus cannot be considered an inner product. In the infinite dimensional case, it is possible for a mapping to satisfy these properties and be considered an inner product, but it is not guaranteed. It depends on the specific mapping and vector space.
 

1. What is an inner product space?

An inner product space is a mathematical concept that consists of a vector space and an inner product, which is a function that takes two vectors and outputs a scalar. This scalar represents the magnitude of the projection of one vector onto the other, and is used to measure the angle between the vectors. Inner product spaces are used in linear algebra and functional analysis to study vector spaces in a geometric way.

2. What is the difference between an inner product space and a normed vector space?

An inner product space is a special type of normed vector space. While both have a notion of length or magnitude for vectors, an inner product space also includes a notion of angle between vectors. This means that inner product spaces have additional structure compared to normed vector spaces, and thus have more properties and theorems that can be applied to them.

3. How can I determine if a given set of vectors form an inner product space?

To determine if a set of vectors form an inner product space, you need to check if the inner product satisfies certain properties. These properties include linearity in the second argument, conjugate symmetry, and positive definiteness. If the inner product satisfies these properties for all vectors in the set, then it forms an inner product space.

4. Can an inner product space have an infinite number of dimensions?

Yes, an inner product space can have an infinite number of dimensions. In fact, many commonly used inner product spaces, such as the space of continuous functions on a closed interval, have infinite dimensions. In these cases, the inner product is defined using integrals rather than dot products, but the properties of linearity, conjugate symmetry, and positive definiteness still hold.

5. What are some real-life applications of inner product spaces?

Inner product spaces have various applications in fields such as physics, engineering, and data analysis. In physics, they are used to model physical systems and describe relationships between quantities, such as force and displacement. In engineering, they are used in signal processing and control systems. In data analysis, inner product spaces are used to represent and analyze data sets, such as images and audio signals.

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