Introducing Inner Product Spaces: Real Impact Examples

[matqkks]

What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of general vectors spaces such as the set of matrices, polynomials, functions?

 

[HallsofIvy]

Actually, you have given the main motivation for inner product spaces: they generalize the Euclidean spaces with dot product. Of course, any finite dimensional space is isomorphic to the Euclidean space of the same dimension so we could just use the "dot product" defined on the Euclidean space. The reason for the more general definitions is to be able to work with infinite dimensional, and in particular, function spaces.

But inner products are also important in adding a topology to vector spaces so that we can talk about "limits" and "continuity". The most general way we can introduce a topology is to add a "distance" or metric function, a function that maps pairs of vectors to a positive real number such that:
i) $d(u, v)\ge 0$ and d(u, v)= 0 if and only if u= v.
ii) $d(u,v)\le d(u, w)+ d(w, v)$ for any vectors, u, v, w.

We interpret d(u, v) as the distance between vectors u and v (a "metric vector space" in which Cauchy sequences converge is called a "Frechet" space). That doesn't really make use of the algebraic properties of a vector space but leads to a "normed space" which has a norm, a function that maps a single vector to a real number such that
i) $|v|\ge 0$ and |v|= 0 if and only if v= 0.
ii) $|u+ v|\le |u|+ |v|$.
iii) $|\alpha v|= |\alpha| |v|$ for any vector v and scalar $\alpha$ ($|\alpha|$ is the usual absolute value of $\alpha$).
and we interpret |v| as the "length" of vector v. Of course, if we have a "length" we have a "distance" and can define d(u, v)= |u- v| so we have all the properties of a "metric vector space" plus additional ones. A normed space in which all Cauchy sequences of vectors converge is called a "Banach space". Other than the Euclidean spaces themselves, the most important example of a Banach space is L₁, the set of all function, f, whose absolute values are Lebesque integrable on a given set, U: $\int_U |f(x)|dx$ is finite and we deifine $|f|= \int_U |f(x)|dx$.

We define an "inner product" on a vector space as a function that maps pairs of vectors to scalars (typically real or complex numbers), satisfying
i) $<i, i>\ge 0$ and <i, i>= 0 if and only if v= 0
ii) $<\alpha u, v>= \alpha<u, v>$ for any two vectors, u and v, and any scalar $\alpha$.
iii)$<u+v, w>= <u, w>+ <v, w>$
Once we have an inner product we can define $|v|= \sqrt{<v, v>}$ and have all of the properties of a normed space (which includes all of the properties of a metric vector space) plus additional properties. An inner product space in whichy all Cauchy sequences of vectors converge is called a "Hilbert space". Other than the Euclidean space themselves, the most important Hilbert space is L₂, the set of all "square integrable" on set U: $\int_U f^2(x)dx$ is finite. One can show that, if both f and g are square integrable on U, then $\int_U f\bar g dx$ is finite and we can define $<f, g>= \int_U f\bar g dx$.

 

[matqkks]

Thanls for the detailed reply.

 

[chiro]

A simple way to think about the motivation of inner product spaces is that it gives a vector space geometry.

With inner product spaces you can do projections which has not only a geometric importance, but an importance in terms of an abstract decomposition.

 

[CellCoree]

One of the most motivating ways to introduce general inner product spaces is by using real world examples that demonstrate the practical applications of this concept. For Euclidean spaces, we can relate the dot product to the angle between vectors, which is a tangible concept that people can easily understand. We can extend this idea to the inner product of general vector spaces by showing how it can be used to measure the similarity or dissimilarity between vectors in various contexts.

For example, in the set of matrices, the inner product can be used to measure the similarity between two matrices, which can be useful in fields such as image processing and data analysis. In polynomials, the inner product can be used to find the best fit curve for a set of data points, which has applications in statistics and engineering. In functions, the inner product can be used to measure the correlation between two functions, which is important in fields such as signal processing and economics.

Furthermore, the concept of orthogonality in inner product spaces can also be applied to real world scenarios. In Euclidean spaces, two vectors are orthogonal if their dot product is zero, and this can be extended to other vector spaces as well. For example, in the set of functions, two orthogonal functions can represent independent variables, which is crucial in fields such as physics and economics.

In summary, by providing real world examples and applications, we can show the real impact of inner product spaces and how they can be used to solve problems and make meaningful connections in various fields of study. This can make the concept more tangible and motivating for students and researchers alike.