# Extend the functional by continuity (Functional analysis)

• mathdunce
In summary, the conversation discusses extending a bounded linear operator T0 from a dense linear subspace E of a normed vector space X to a Banach space Y without increasing its norm. The proposed solution involves showing that if (xn) is a convergent sequence in E, then T0(xn) is a Cauchy sequence, defining f(x) as the limit of this sequence, and using the fact that T0 is uniformly continuous to show that f(x) is a well-defined bounded linear function.
mathdunce

## Homework Statement

Let E be a dense linear subspace of a normed vector space X, and let Y be a
Banach space. Suppose T0 $\in$ £(E, Y) is a bounded linear operator from E to Y.
Show that T0 can be extended to T$\in$ £(E, Y) (by continuity) without increasing its norm.

## The Attempt at a Solution

Someone kindly gave me a hint. I am trying to work out the details. The deadline is approaching. So I put the question here just in case. Thanks.
For this particular problem you want to show that if (xn) converges to x then T0(xn) is a Cauchy sequence and then define f(x) as the limit of the sequence. Finally you need to show that the map is a well-defined bounded linear function.

Use that T0 is uniform continuous.

Thank you!

## 1. What is the concept of continuity in functional analysis?

The concept of continuity in functional analysis refers to the smooth and consistent behavior of a function as its input values change. In other words, a continuous function is one that has no sudden jumps or breaks in its graph.

## 2. How does continuity affect the behavior of a function?

Continuity ensures that small changes in the input of a function result in small changes in the output. This is important in functional analysis as it allows for the study of the behavior of a function as its input approaches a particular value or as it changes over a certain interval.

## 3. How is continuity related to differentiability in functional analysis?

In functional analysis, differentiability is a stronger condition than continuity. A function is differentiable if its derivative exists at every point, and a differentiable function is also continuous. However, there are continuous functions that are not differentiable, which highlights the importance of studying continuity separately.

## 4. Can a function be continuous at one point but not at others?

Yes, it is possible for a function to be continuous at one point but not at others. This is known as a point of discontinuity. A function can have different types of discontinuities, such as jump discontinuities, removable discontinuities, and essential discontinuities.

## 5. How is the concept of continuity used in practical applications of functional analysis?

The concept of continuity is used in various practical applications of functional analysis, such as in optimization problems, differential equations, and signal processing. It allows for the study of the behavior of a function over a certain domain, which can provide valuable insights into real-world problems and applications.

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