/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Let \(p\) be a sublinear functio... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 9

Let \(p\) be a sublinear functional on a real vector space \(X\). Let \(f\) be defined on \(Z=\left\\{x \in X \mid x=\alpha x_{0}, \alpha \in \mathbf{R}\right\\}\) by \(f(x)=\alpha p\left(x_{0}\right)\) with fixed \(x_{0} \in X\). Show that \(f\) is a linear functional on \(Z\) satisfying \(f(x) \leqq p(x)\).

Short Answer

Expert verified
The functional \( f(x) = \alpha p(x_0) \) is linear and satisfies \( f(x) \leq p(x) \) on \( Z \).

Step by step solution

01

Understand the Problem

We need to show that the functional \( f \) defined on the set \( Z \) is linear and satisfies \( f(x) \leq p(x) \) for all \( x \in Z \). The set \( Z \) consists of all scalar multiples of a fixed vector \( x_0 \).
02

Define a Sublinear Functional

Recall that a sublinear functional \( p \) satisfies:1. \( p(x+y) \leq p(x) + p(y) \) for all \( x, y \in X \) (subadditivity), and2. \( p(\alpha x) = \alpha p(x) \) for all \( \alpha \geq 0 \) and \( x \in X \) (positive homogeneity).
03

Check Linearity of f

To show \( f \) is linear, we need to prove it satisfies additivity and homogeneity. For any \( x = \alpha x_0 \) and \( y = \beta x_0 \) in \( Z \):- Additivity: \( f(x + y) = f((\alpha + \beta)x_0) = (\alpha + \beta)p(x_0) = \alpha p(x_0) + \beta p(x_0) = f(x) + f(y) \).- Homogeneity: For \( \gamma \in \mathbb{R} \), \( f(\gamma x) = f(\gamma \alpha x_0) = \gamma \alpha p(x_0) = \gamma f(x) \).
04

Show f(x) 鈮� p(x) for all x in Z

Given \( x \in Z \), \( x = \alpha x_0 \), by definition \( f(x) = \alpha p(x_0) \). We want to show \( f(x) \leq p(x) \):Since \( p \) is positive homogeneous, \( p(x) = p(\alpha x_0) = \alpha p(x_0) \). Thus, \( f(x) = \alpha p(x_0) \leq p(x) = \alpha p(x_0) \), which is trivially satisfied.
05

Conclusion

Since both linearity and the inequality \( f(x) \leq p(x) \) are satisfied across all elements in \( Z \), we can conclude that \( f \) is a linear functional satisfying the given condition. So, the proof is complete.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sublinear Functional
A sublinear functional is a special kind of functional on a vector space that satisfies two important properties: subadditivity and positive homogeneity.
It helps measure an element of a vector space in a way that鈥檚 not necessarily linear, but still maintains some structure.

To understand the core properties of a sublinear functional, consider:
  • Subadditivity: For any vectors \( x \) and \( y \) in a vector space \( X \), a sublinear functional \( p \) will satisfy \( p(x + y) \leq p(x) + p(y) \).
    This means the functional value of a sum is never greater than the sum of the individual functional values.
  • Positive Homogeneity: For any non-negative scalar \( \alpha \) and any vector \( x \) in \( X \), it holds that \( p(\alpha x) = \alpha p(x) \).
    This property implies that scaling a vector by a positive real number scales its functional value by the same factor.
In the context of the problem, the functional \( p \) helps in defining another functional \( f \) and is used to prove properties about \( f \).
Understanding these properties is key to proving whether \( f \) meets the criteria for being linear.
Vector Space
A vector space is a fundamental concept in mathematics, particularly in linear algebra and functional analysis.
It consists of a collection of objects called vectors, which can be added together and multiplied by scalars (real numbers).

The defining characteristics of a vector space include:
  • Closure under Addition: The sum of any two vectors \( u \) and \( v \) in the vector space \( X \) will also be a vector in \( X \).
  • Closure under Scalar Multiplication: Any scalar multiplication of a vector \( v \) by a real number \( \alpha \) will result in a vector that is still within \( X \).
  • Containment of the Zero Vector: There is always a zero vector \( 0 \) in the space, fulfilling the property that adding it to any vector does not change the original vector (i.e., \( v + 0 = v \)).
  • Associativity and Commutativity of Addition: Adding vectors is associative (\((u + v) + w = u + (v + w)\)) and commutative (\(u + v = v + u\)).
  • Distributive and Associative Properties of Scalar Multiplication: Scalar operations distribute over vector addition, and scalar multiplication is associative.
In the problem, the set \( Z \) is a special subset of a vector space \( X \), consisting of all scalar multiples of a fixed vector \( x_0 \).
This subset forms a kind of 'line' through the vector space, which is itself a type of vector space.
Linear Functional
A linear functional is a particular type of function that maps elements from a vector space to the real numbers.
As the name suggests, it retains linear properties: additivity and homogeneity.

Here's how these two properties apply:
  • Additivity: For vectors \( x \) and \( y \) in the vector space, a linear functional \( f \) satisfies \( f(x + y) = f(x) + f(y) \).
    This means the function respects the operation of vector addition.
  • Homogeneity: For any vector \( x \) in the vector space and any scalar \( \lambda \), \( f(\lambda x) = \lambda f(x) \).
    This property ensures that scaling the vector by a scalar results in scaling the function output by the same factor.
The task is to show \( f \) is a linear functional on a defined subset \( Z \), with the additional condition that \( f(x) \leq p(x) \) for all \( x \in Z \).
Such demonstrations often involve confirming that these two core properties hold true for any combinations of vectors and scalars in the subset, using the constraints the subset provides.

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Most popular questions from this chapter

(Ces脿ro's \(\boldsymbol{C}_{\mathrm{k}}\)-method) Given \(\left(\xi_{n}\right)\), let \(\sigma_{n}^{(0)}=\xi_{n}\) and $$ \sigma_{n}^{(k)}=\sigma_{0}^{(k-1)}+\sigma_{1}^{(k-1)}+\cdots+\sigma_{n}^{(k-1)} \quad(k \geqq 1, n=0,1,2, \cdots) $$ If for a fixed \(k \in \mathbf{N}\) we have \(\eta_{n}^{(k)}=\sigma_{n}^{(k)} /\left(\begin{array}{c}n+k \\ k\end{array}\right) \longrightarrow \eta\), we say that \(\left(\xi_{n}\right)\) is \(C_{k}\)-summable and has the \(C_{k}\)-limit \(\eta\). Show that the method has the advantage that \(\sigma_{n}^{(k)}\) can be represented in terms of the \(\xi\) 's in a very simple fashion, namely $$ \sigma_{n}^{(k)}=\sum_{\nu=0}^{n}\left(\begin{array}{c} n+k-1-v \\ k-1 \end{array}\right) \xi_{v} $$.

Show that any closed subspace \(Y\) of a normed space \(X\) contains the limits of all weakly convergent sequences of its elements.

If a subadditive functional \(p\) on a normed space \(X\) is continuous at 0 and \(p(0)=0\), show that \(p\) is continuous for all \(x \in X\).

Let \(X\) and \(Y\) be Banach spaces and \(T: X \longrightarrow Y\) an injective bounded linear operator. Show that \(T^{-1}: \Re(T) \longrightarrow X\) is bounded if and only if \(\mathscr{R}(T)\) is closed in \(Y\).

Let \(X\) and \(Y\) be normed spaces and \(X\) compact. If \(T: X \longrightarrow Y\) is a bijective closed linear operator, show that \(T^{-1}\) is bounded.
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