# Consider C[0,1] with sup metric.

• coverband
In summary, C[0,1] with sup metric is a set of continuous functions defined on the interval [0,1] that are measured using the supremum norm. The supremum norm is defined as the maximum absolute value of a function over a given interval and is useful for measuring the distance between functions. It is commonly used in mathematics and science, as well as in engineering and physics. Other important metrics used in C[0,1] include the Lp norm and the uniform norm.
coverband
jjjj

Okay, I'm considering it!

(How long? I have a class to go to soon!)

For those who are wondering, "C[0, 1]" is the set of functions, f(x), that are continuous on the interval [0,1].

The "sup metric", also called "supremum metric" measures the "distance" between functions by the largest difference between values: max |f(x)- g(x)| over all x between 0 and 1. Notice that since f and g are continuous so is f- g and so the maximum over the closed and bounded interval [0, 1] does exist.

Of course, one would hope that there is some "problem" associated with this.

## 1. What is C[0,1] with sup metric?

C[0,1] with sup metric refers to the set of all continuous functions defined on the interval [0,1], with the metric of supremum norm. This means that the distance between two functions is measured by finding the maximum value of the absolute difference between the two functions over the interval [0,1].

## 2. How is the supremum norm defined?

The supremum norm, denoted as ||f||sup, is defined as the maximum absolute value of a function f(x) over a given interval. In the case of C[0,1] with sup metric, the interval is [0,1]. Mathematically, it can be written as ||f||sup = sup{|f(x)| : x ∈ [0,1]}.

## 3. What is the significance of using the supremum norm in this context?

The supremum norm is a useful metric for measuring the distance between functions because it takes into account the entire range of values of a function, rather than just a single point. This makes it a stronger metric than other norms, such as the Lp norm, which only consider a finite number of points.

## 4. How is the sup metric used in real-world applications?

The sup metric is commonly used in various fields of mathematics and science, such as functional analysis, optimization, and numerical analysis. It is also used in engineering and physics, particularly in the study of differential equations and control systems.

## 5. Are there any other important metrics used in C[0,1]?

Yes, besides the sup metric, there are other commonly used metrics in C[0,1], such as the Lp norm and the uniform norm. Each metric has its own applications and properties, and the choice of which one to use depends on the specific problem at hand.

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