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