Space l_infty(R) with sup norm

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Your Name]In summary, we can prove that the space l_\infty(R) of bounded sequences with the sup norm is not an inner product space by showing that the sup norm does not satisfy the linearity and triangle inequality properties, which are essential for defining an inner product and a norm. This highlights the importance of these properties in determining whether a space is an inner product space or not.
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Homework Statement


Prove that the space l_\infty (R) of bounded sequences with the sup norm (ie x=(x_n) in l_\infty (R) )is not a inner product space.


The Attempt at a Solution



Using definition of parallelogram ||x+y||^2+||x-y||^2=2(||x||^2+||y||^2) (1)

Consider x_n=1^-n and y_n=4-1/n then

||x||_\infty=sup|x_n| = 1 where n=1 and |y||_\infty=sup|y_n|=4 where n=\infty

||x+y||_\infty=|1+4|=5 where I chose n=1 for x_n and n=\infty for y_n so that we get the maximum value.

||x-y||_\infty = |0-4|=4 where I chose n=\infty for x_n and n=1 for y_n to get a maximum

||x+y||^2_\infty+||x-y||^2_\infty=41

THis is not equal to the RHS 2(||x||^2+||y||^2)= 34

implies the space l_\infty(R) is not an inner product space..?
 
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Thank you for your post. I would like to provide a more detailed explanation to show why l_\infty(R) is not an inner product space.

Firstly, an inner product space is a vector space equipped with an inner product, which is a function that takes in two vectors and returns a scalar value. This inner product must satisfy certain properties, such as linearity, symmetry, and positive definiteness. In the case of l_\infty(R), the sup norm ||x||_\infty does not satisfy the linearity property, as shown in your proof. This means that it cannot be considered as an inner product.

Furthermore, in an inner product space, the inner product can be used to define a norm, which is a function that maps a vector to a non-negative scalar value. The norm must also satisfy certain properties, such as non-negativity, homogeneity, and the triangle inequality. However, in l_\infty(R), the sup norm does not satisfy the triangle inequality, as shown in your proof. This means that it cannot be used to define a norm, which is a crucial property of an inner product space.

In conclusion, l_\infty(R) cannot be considered as an inner product space due to the violation of fundamental properties of an inner product and a norm. Thank you for bringing up this interesting topic for discussion.
 

What is "Space l_infty(R) with sup norm"?

"Space l_infty(R) with sup norm" refers to a mathematical concept in functional analysis that represents a set of real-valued functions with a specific type of norm, known as the sup norm or the infinity norm. It is denoted by l_infty(R) and is a subset of the space of bounded functions on the real line.

What is a norm?

In mathematics, a norm is a function that assigns a positive length or size to a mathematical object, such as a vector or a function. It is a way of measuring the magnitude or size of an object in a mathematical space. In the case of "Space l_infty(R) with sup norm", the norm used is the sup norm, which is defined as the supremum or the least upper bound of the absolute values of a function.

What is the significance of "Space l_infty(R) with sup norm" in functional analysis?

"Space l_infty(R) with sup norm" is important in functional analysis because it allows us to study the properties of bounded functions on the real line. This space is complete, meaning that all Cauchy sequences converge within the space, and is also a Banach space, which means it is a complete normed vector space. It is also useful in various applications, such as in the study of differential equations and Fourier analysis.

How is "Space l_infty(R) with sup norm" different from other function spaces?

One key difference between "Space l_infty(R) with sup norm" and other function spaces is that it consists of only bounded functions. This means that the functions in this space do not have to approach a limit or tend to infinity as the input variable increases. Additionally, the sup norm used in this space is different from other norms, such as the Lp norms, which are based on integrals rather than the supremum of a function.

What are some practical applications of "Space l_infty(R) with sup norm"?

"Space l_infty(R) with sup norm" has various practical applications, especially in the fields of engineering and physics. For example, it can be used to analyze the behavior of signals in electrical engineering or to study the stability of systems in control theory. It is also used in the analysis of time series data and in the study of probability theory. Additionally, the concept of sup norm is used in optimization problems in economics and finance.

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