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Homework Statement
For the Banach space [tex]X = C[0,1][/tex] with the supremum norm, fix
an element [tex]g \in X[/tex] and define a map [tex]\varphi_g : X \to \mathbb{C}[/tex]
by
[tex]
\begin{align*}
\varphi_g(h) := \int^1_0 g(t) h(t) dt, \qquad h \in X
\end{align*}
[/tex]
Define [tex]W := \{ \varphi_g | g \in X \}[/tex].
Prove that [tex]\varphi_g \in X^*[/tex] and calculate
[tex]||\varphi_g||_{X^*}[/tex].
Homework Equations
The Attempt at a Solution
The supremum norm for a function [tex]f \in C[0,1][/tex] is [tex]\displaystyle
||f|| = \sup_{x \in [0,1]} |f(x)|[/tex].
For [tex]h_1,h_2 \in X[/tex] and [tex]\lambda \in \mathbb{C}[/tex]
[tex]
\begin{align*}
\varphi_g(h_1 + h_2) =& \int^1_0 g(t) (h_1(t) + h_2(t)) dt =
\int^1_0 g(t) h_1(t) dt + \int^1_0 g(t) h_2(t) dt = \varphi_g(h_1) +
\varphi_g(h_2) \\
\varphi_g(\lambda h_1) =& \int^1_0 \lambda g(t) h_1(t) dt = \lambda
\int^1_0 g(t) h_1(t) dt
\end{align*}
[/tex]
So [tex]\varphi_g[/tex] is linear functional.
For any [tex]h \in X[/tex], [tex]\varphi_g[/tex] is continuous if [tex]\forall \varepsilon
> 0[/tex] [tex]\exists \delta > 0[/tex] such that any [tex]h' \in X[/tex] satisfies
[tex]
\begin{align*}
d(h,h') < \delta \implies d(\varphi_g(h), \varphi_g(h')) <
\varepsilon \text{.}
\end{align*}
[/tex]
The distance function is [tex]\displaystyle d(h,h') = ||h - h'|| =
\sup_{x \in [0,1]} |h(t) - h'(t)|[/tex].
[tex]
\begin{align*}
d(\varphi_g(h), \varphi_g(h')) =& |\int^1_0 g(t) h(t) dt -
\int^1_0 g(t) h'(t) dt | = |\int^1_0 g(t) h(t) - g(t) h'(t) dt | \\
=& |\int^1_0 g(t) (h(t) - h'(t)) dt| \leq \int^1_0 |g(t) (h(t) - h'(t))| dt \\
\leq & \int^1_0 |g(t)| |(h(t) - h'(t))| dt \leq \sup_{t \in
[0,1]} |(h(t) - h'(t))| \int^1_0 |g(t)| dt \\
=& d(h,h') \int^1_0 |g(t)| dt
\end{align*}
[/tex]
For any [tex]\varepsilon > 0[/tex], we get [tex]\displaystyle d(h,h') \int^1_0
|g(t)| dt < \varepsilon[/tex] when [tex]\displaystyle d(h,h') <
\frac{\varepsilon}{\displaystyle \int^1_0 |g(t)| dt} = \delta[/tex].
Thus [tex]\varphi_g[/tex] is continuous. Are the above proofs correct?
The formula for [tex]||\varphi_g||_{X^*}[/tex] is
[tex]
\begin{align*}
||\varphi_g||_{X^*} :=& \sup \{ |\varphi_g(h)| | \norm{h} = 1
\} \\
=& \sup \left\{ |\int^1_0 g(t)h(t) dt| \bigg| \sup_{t \in [0,1]}
|h| = 1 \right\}
\end{align*}
[/tex]
How can I find the value of this?
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