When is a metric given by a norm

IP4 ,. A vector space with a specified inner product is called an inner product space. One of the most basic examples, in the case of a finite-dimensional vector space, is given by the following procedure.

Let and be elements vectors of some -dimensional real vector space , with respective components and in some basis. Then we can set. To make things even easier to visualize, let us set , so that we are dealing with vectors which we can now think of as quantities with magnitude and direction in the plane.

These two vectors are perpendicular, or orthogonal. Computing the inner product we discussed earlier, we have. We say, therefore, that two vectors are orthogonal when their inner product is zero. As we have mentioned earlier, we can extend this to cases where our geometric intuition may no longer be as useful to us. We set our inner product to be. As an example, let and. But if we take the inner product, we will see that.

Hence we see that and are orthogonal. Similarly, we have. We have discussed this in more detail in Some Basics of Fourier Analysis.

We have also seen in that post that orthogonality plays a big role in the subject of Fourier analysis. Just as a norm always induces a metric, an inner product also induces a norm, and by extension also a metric. In other words, an inner product space is also a normed space, and also a metric space.

The norm is given in terms of the inner product by the following expression:. Just as with the norm and the metric, although an inner product always induces a norm, not every norm is induced by an inner product. There is one more concept I want to discuss in this post. In Valuations and Completions , we discussed Cauchy sequences and completions. Those concepts still carry on here, because they are actually part of the study of metric spaces in fact, the valuations discussed in that post actually serve as a metric on the fields that were discussed, showing how in number theory the concept of metric and metric spaces still make an appearance.

If every Cauchy sequence in a metric space converges to an element in , then we say that is a complete metric space. Since normed spaces and inner product spaces are also metric spaces, the notion of a complete metric space still makes sense, and we have special names for them. A normed space which is also a complete metric space is called a Banach space , while an inner product space which is also a complete metric space is called a Hilbert space.

Finite-dimensional vector spaces over the real or complex numbers are always complete, and therefore we only really need the distinction when we are dealing with infinite dimensional vector spaces. Banach spaces and Hilbert spaces are important in quantum mechanics. We recall in Some Basics of Quantum Mechanics that the possible states of a system in quantum mechanics form a vector space.

Meanwhile, Banach spaces often arise when studying operators , which correspond to observables such as position and momentum. Of course the states form Banach spaces too, since all Hilbert spaces are Banach spaces, but there is much motivation to study the Banach spaces formed by the operators as well instead of just that formed by the states. This is an important aspect of the more mathematically involved treatments of quantum mechanics.

Topological Vector Space on Wikipedia. Complete Metric Space on Wikipedia. Pingback: Adeles and Ideles Theories and Theorems. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Search for:. Metric We start with the concept of a metric. Munkres: A metric on a set is a function having the following properties: 1 for all ; equality holds if and only if.

We quote from the same book another important definition: Given a metric d on X, the number is often called the distance between and in the metric. Finally, once more from the same book, we have the definition of the metric topology: If is a metric on the set , then the collection of all -balls , for and , is a basis for a topology on , called the metric topology induced by.

For instance, we may simply put if if. Norm Now we move on to vector spaces we will consider in this post only vector spaces over the real or complex numbers , and some mathematical concepts that we can associate with them, as suggested in the beginning of this post. The metric is given in terms of the norm by the following equation: However, not all metrics come from a norm.

Inner Product Next we discuss the inner product. Again, for the technical definition we quote from the book of Kreyszig: With every pair of vectors and there is associated a scalar which is written and is called the inner product of and , such that for all vectors , , and scalars we have IPl IP2 IP3 IP4 , A vector space with a specified inner product is called an inner product space.

Similarly, we have and and are also orthogonal. The other question, which I would summarize as "which came first? Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Difference between metric and norm made concrete: The case of Euclid Ask Question. Asked 10 years, 6 months ago. Active 4 years, 6 months ago. Viewed 32k times.

Wikipedia says: A vector can be described as a directed line segment from the origin of the Euclidean space vector tail , to a point in that space vector tip. I would very much appreciate it if somebody could clear the haze. You can define the norm in terms of the metric, or you can define the metric in terms of the norm. It doesn't matter which order you do it in. Perhaps my confusion stems from the fact that on the one hand, as you say, you can do it either way, on the other hand there is this hierarchy between metric spaces and normed spaces every normed space is a metric space but not the other way round.

It seems like a contradiction to me?!? There is no contradiction here. Add a comment. Active Oldest Votes. Gerry Myerson Gerry Myerson k 11 11 gold badges silver badges bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.

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