# Is every norm preserved under a unitary map?

• Bipolarity
In summary, unitary operators are norm preserving and inner product preserving, but not all norms are preserved.
Bipolarity
I am a bit confused, so this question may not make much sense.

A unitary operator from one vector space to another is one whose inverse and Hermitian transpose are identical.

It can be proved that unitary operators are norm preserving and inner product preserving. Which raises the question, which norm/inner product does this refer to, since the unitarity of the map was not defined with respect to an inner product?

In particular, I ask this question in application to signal processing.
Consider the space of functions that are integrable and square-integrable from -∞ to ∞. This space is a vector space, and also an inner product space under the standard function inner product. The norm induced by this inner product is the ##L^{2}## norm. It can be shown to be preserved under the Fourier transform, a result known as Parseval's theorem.

But does the Fourier transform preserve the ##L^{1}## norm, which does not appear to be induced by any inner product that I know of?

Thanks!

BiP

Fourier transform of L1 function is bounded, with the bound ≤ norm of the original function. In other words the Fourier transform of an L1 function is L∞.

Bipolarity said:
A unitary operator from one vector space to another is one whose inverse and Hermitian transpose are identical.
That's one way to define a unitary operator. Note well: It doesn't make sense to talk about UU* without having some concept of an inner product. Another way to define a unitary operator is as a surjective operator that preserves the inner product. The two definitions are equivalent.

It can be proved that unitary operators are norm preserving and inner product preserving. Which raises the question, which norm/inner product does this refer to, since the unitarity of the map was not defined with respect to an inner product?
Unitary operators preserve the inner product, which means they preserve the norm induced by that inner product. They do not preserve all norms. A simple example: Consider ℝ2. Rotate ##\hat x## by 45 degrees to ##\frac{\sqrt{2}}2 (\hat x + \hat y)##. The Euclidean norm (L2 norm) of that rotated unit vector is still one, but the taxicab norm (L1 norm) has changed from one to √2, and the max norm (L norm) has changed from one to √2/2.

## 1. What is a norm?

A norm is a mathematical concept that measures the size or magnitude of a vector, matrix, or other mathematical object. It is a way to quantify the distance between two points or the length of a vector. Examples of norms include the Euclidean norm and the Frobenius norm.

## 2. What does it mean for a norm to be preserved under a unitary map?

A unitary map is a linear transformation that preserves the inner product and the norm of a vector. This means that the norm of a vector before the transformation will be equal to the norm of the vector after the transformation. In other words, the size or magnitude of the vector remains unchanged after the unitary map.

## 3. Why is it important for norms to be preserved under a unitary map?

Preserving norms under a unitary map is important in many areas of mathematics, physics, and engineering. It allows us to analyze and understand the behavior of systems that undergo transformations without changing their overall structure. This is especially useful in quantum mechanics, where unitary maps are used to represent physical transformations.

## 4. Is every norm preserved under a unitary map?

No, not every norm is preserved under a unitary map. Only unitarily invariant norms, such as the Euclidean norm and the Frobenius norm, are preserved. Other norms, such as the maximum norm or the L1 norm, may change under a unitary map.

## 5. How can we determine if a norm is preserved under a unitary map?

To determine if a norm is preserved under a unitary map, we can use the properties of unitary matrices. If a norm satisfies the condition ||Ux|| = ||x|| for all vectors x and unitary matrices U, then it is considered unitarily invariant and will be preserved under a unitary map.

Replies
17
Views
1K
Replies
8
Views
2K
Replies
2
Views
3K
Replies
15
Views
10K
Replies
1
Views
2K
Replies
4
Views
4K
Replies
4
Views
5K
Replies
2
Views
2K
Replies
11
Views
1K
Replies
43
Views
6K