Bounded Linear Functional and Hahn-Banach Theorem
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Questions and Answers

Which statement correctly describes the relationship between ker(f) and S_r(y0)?

  • ker(f) intersects S_r(y0). (correct)
  • S_r(y0) is entirely contained within ker(f).
  • ker(f) and S_r(y0) are disjoint.
  • S_r(y0) is a subset of ker(f).

    • What can be concluded about the function f based on the defined neighborhood S_r(y0)?

  • f is bounded by 1 in S_r(y0). (correct)
  • f is unbounded in S_r(y0).
  • f can take values greater than or equal to 1 in S_r(y0).
  • f must be constant in S_r(y0).

    • What does the inequality f(x) < 1 indicate for the boundary case presented with x = 0?

  • f(0) must be undefined.
  • f(0) equals any positive scalar.
  • f(0) equals zero. (correct)
  • f(0) can take any negative value.

    • In the context of the content, what implication does the established result f(x) ≤ (2/r)x have?

    <p>f is a continuous linear functional.</p> Signup and view all the answers

    What does the remark about Theorem 3.1.6 suggest regarding linear transformations?

    <p>The theorem holds only for specific types of normed spaces.</p> Signup and view all the answers

    What does the notation B(N, N') signify in the context provided?

    <p>The space of continuous linear transformations from N to N'.</p> Signup and view all the answers

    What is the condition for f defined on N as stated in the proof?

    <p>f is a bounded linear functional.</p> Signup and view all the answers

    Which of the following attributes does the term 'dual space' refer to in the content?

    <p>It describes the space of continuous linear transformations.</p> Signup and view all the answers

    Which statement accurately describes the linearity of the functional f?

    <p>f(y + y') equals f(y) plus f(y') for any y, y' in M0.</p> Signup and view all the answers

    What condition indicates that f is bounded?

    <p>f(y) ≤ 1/d for all non-zero y in M0.</p> Signup and view all the answers

    In case 1 where α ≠ 0, how does f relate to the distance d?

    <p>f(y) is less than or equal to y/d.</p> Signup and view all the answers

    How is f defined on M0?

    <p>f(m + αx0) = α for all m in M and α in K.</p> Signup and view all the answers

    Which conclusion can be drawn if α = 0?

    <p>f is still bounded even if y is not zero.</p> Signup and view all the answers

    What property of the functional f is proven in the content?

    <p>f is linear and continuous.</p> Signup and view all the answers

    What does the supremum condition f ≤ 1/d imply?

    <p>f remains below a threshold determined by the distance d.</p> Signup and view all the answers

    What does the continuity of f suggest about its behavior?

    <p>f's value can be predicted based on the inputs in M0.</p> Signup and view all the answers

    What does the expression $f_0(y) \leq y$ imply for all $y \in M_0$?

    <p>$f_0$ is bounded above by 1.</p> Signup and view all the answers

    Which property of $g$ and $h$ is demonstrated in the proof concerning $f$ as a linear functional?

    <p>$g$ and $h$ are both bounded linear functionals.</p> Signup and view all the answers

    How is the supremum of $f_0$ expressed mathematically in the context provided?

    <p>$f_0 = \sup f(y): y \in M_0, y \neq 0$</p> Signup and view all the answers

    What condition must $ alpha$ satisfy to maintain the inequality involving $f_0$?

    <p>$\alpha$ must be less than 0.</p> Signup and view all the answers

    Which of the following statements correctly summarizes the relationship between $f$ and $f_0$?

    <p>$f_0$ is an extension of $f$, evaluated at $M_0$.</p> Signup and view all the answers

    What is the result when $f$ is proven to equal 1?

    <p>$f_0$ is confirmed to equal 1.</p> Signup and view all the answers

    What is the relationship between the functional f0 and the closed linear subspace M?

    <p>f0 is a linear functional on M.</p> Signup and view all the answers

    In the context given, what can be inferred about the boundedness of the functionals $g$ and $h$?

    <p>$g$ and $h$ are bounded linear functionals on $M$.</p> Signup and view all the answers

    Which property characterizes the relationship f(x + y) = f(x) + f(y) in terms of linearity?

    <p>It suggests that $f$ is a linear functional.</p> Signup and view all the answers

    Which statement correctly describes the nature of the mapping T?

    <p>T is a continuous linear transformation from N to N/M.</p> Signup and view all the answers

    What must hold true about the vector x0 in relation to the subspace M?

    <p>x0 cannot be an element of M.</p> Signup and view all the answers

    According to the proof, which property of g assures that g(x0 + M) is non-zero?

    <p>g is defined such that g(x0 + M) equals x0 + M.</p> Signup and view all the answers

    How is the norm on the quotient space N/M defined?

    <p>It is determined by the infimum of distances to M.</p> Signup and view all the answers

    Which of the following correctly states a characteristic of functional f0?

    <p>f0 is linear with respect to its inputs from M.</p> Signup and view all the answers

    What conclusion can be drawn about the zero vector in the context of the quotient space N/M?

    <p>The zero vector in N is mapped to the zero vector in N/M.</p> Signup and view all the answers

    What does Theorem 3.4.5 contribute to the proof presented?

    <p>It guarantees the existence of functional f0.</p> Signup and view all the answers

    What is the relationship established between h(x) and g(ix)?

    <p>h(x) = -g(ix)</p> Signup and view all the answers

    For f0 to be considered functional, what property must it satisfy?

    <p>It must be linear.</p> Signup and view all the answers

    What does the expression f0(x + y) signify in the context of linearity?

    <p>The sum of functional values at x and y.</p> Signup and view all the answers

    What is the form of f0(x) as defined in the document?

    <p>f0(x) = g0(x) - ig0(ix)</p> Signup and view all the answers

    Given the linearity of f0, what condition must hold for any scalar a?

    <p>f0(ax) = af0(x)</p> Signup and view all the answers

    What is the significance of the term i2 used in the context of f0?

    <p>It relates to complex number properties.</p> Signup and view all the answers

    What conclusion can be drawn about f0 given that it is established as linear?

    <p>f0 satisfies the linearity condition for all scalars.</p> Signup and view all the answers

    What must be true for the extended function g0 defined on M0?

    <p>g0 extends g to a larger domain.</p> Signup and view all the answers

    Study Notes

    Bounded Linear Functional

    • For a normed linear space, a bounded linear functional is a linear transformation that is also bounded, meaning there exists a constant C such that for all vectors x in the space:

    ||f(x)|| ≤ C || x ||

    • The Hahn-Banach Theorem states that every bounded linear functional defined on a subspace can be extended to a bounded linear functional on the entire space without increasing the norm.
    • This means that the bounded linear functional can be applied to any vector in the entire space, not just those within the subspace.
    • **The text proves the theorem with the construction of a new linear functional. This functional is defined on the entire space (M0) and is proven to be linear and bounded. The construction ensures that the functional and its norm are the same on the subspace M as the original functional f.
    • Claim: For any vector x in the subspace S, the norm of the linear transformation applied to x is always less than or equal to the norm of x multiplied by a constant.
    • If this claim is true, then the functional f is bounded.

    Conjugate Space/Dual Space

    • It is essentially the space consisting of all bounded linear functionals defined on the normed space. This space is also denoted as N*.
    • The text defines a functional f on M0 using the identity:

    f(m+αx0) = α Where m is an element of the subspace (M), α is a scalar, and x0 is a point outside of M.

    • The text then proceeds to prove that this functional is indeed linear and bounded.
    • It also proves the functional is a bounded linear functional by constructing a sequence that converges to the infimum of the difference between elements in M and x0.
    • This allows to establish that the norm of the functional f is less than or equal to 1/d (where d is the infimum).

    Hahn-Banach Theorem

    • This theorem states that if M is a closed linear subspace of the normed space N, and x0 is a point in N but not in M, then a functional f can be defined on N with the following properties:
    • f(M) = 0: The functional maps all elements of M to 0.
    • f(x0) ≠ 0: The functional maps x0 to a non-zero value.
    • The text demonstrates the theorem with two proofs:
      • One is based on the previously stated Hahn-Banach theorem, showcasing how it can be used to prove this particular case.
      • The other proof utilizes the concept of a quotient space (N/M) and continuous linear transformations, demonstrating how these concepts can be used to define the functional f.
    • **The proof constructs a functional f0 using a quotient space that maps all elements of the subspace to 0 and x0 to a non-zero value. Since the norm of g is 1, the norm of f0 is also shown to be 1, satisfying the theorem's prerequisites. **

    Extensions of Bounded Functionals

    • The text extends the Hahn-Banach Theorem to complex normed spaces by introducing the concept of a "real part" (g) and an "imaginary part" (h) for the functional.
    • It demonstrates that g and h are bounded linear functionals proving that f0 is also a bounded linear functional in N.
    • This extension is crucially important for the use of the Hahn-Banach theorem in complex analysis and quantum mechanics.

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    Description

    Explore the concepts of bounded linear functionals and the Hahn-Banach Theorem in normed linear spaces. This quiz covers the definition, properties, and the process of extending bounded linear functionals from subspaces to the entire space, ensuring the preservation of norms. Test your understanding of these fundamental concepts in functional analysis.

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