linear functionals, closed sets, convergence

**marianne** · June 21st 2008, 12:16 AM

This is the last round of questions, I promise.

(1) Identify the space of all continuous linear functionals on the space $c_0$ .

(2) $X=(C([0, 2], \mathbb{R}), || . ||_2)$ ,
$Y=\{f \in C([0, 2], \mathbb{R}: f(1-x)=f(1+x), x \in [0,1]\}$
Is Y closed vector space in X? What is the complement of Y?

(3) Prove that the series
$\sum_{k=1}^{\infty}x_k b_k$ converge for every vector
$x=(x_k) \in l^p, 1<p<\infty$ if and only if $b=(b_k) \in l^q, 1/p +1/q=1$ .

These are the last ones I don't know what to do with, and if anyone has the time and is willing to help, I would be really grateful.

**Opalg** · June 21st 2008, 01:31 PM

Originally Posted by marianne

(1) Identify the space of all continuous linear functionals on the space $c_0$ .

The dual space of $c_0$ is (isometrically isomorphic to) $l^1$ . If $x=(x_n)\in c_0$ and $y=(y_n)\in l^1$ then the value of the functional y at the point x is $\sum x_ny_n$ . This is a very standard result which you should be able to find in any relevant textbook.

Originally Posted by marianne

(2) $X=(C([0, 2], \mathbb{R}), || . ||_2)$ ,
$Y=\{f \in C([0, 2], \mathbb{R}): f(1-x)=f(1+x), x \in [0,1]\}$
Is Y closed vector space in X? What is the complement of Y?

This is more interesting. Take the second question first. If g(x) is in the orthogonal complement of Y, then $\int_0^2f(x)g(x)\,dx=0$ for all f in Y. But $\int_0^2f(x)g(x)\,dx = \int_0^1f(x)g(x)\,dx + \int_1^2f(x)g(x)\,dx$ . If you change variables in the two integrals, putting y=1-x in the first and y=1+x in the second, then you find that $\int_0^1f(1-y)\bigl(g(1-y)+g(1+y)\bigr)dy=0$ (remembering that f(1+y)=f(1-y)). This holds for all continuous functions on [0,1], so you conclude that g(1+y)=–g(1-y) for all y in [0,1].

Thus the orthogonal complement of Y is $Z=\{g \in C([0, 2], \mathbb{R}): g(1-x)=-g(1+x), x \in [0,1]\}.$ If you repeat the same argument on the subspace Z, you will find that its orthogonal complement is Y. Therefore Y is equal to its second orthogonal complement, which is closed in X. Hence Y is closed.

Originally Posted by marianne

(3) Prove that the series
$\sum_{k=1}^{\infty}x_k b_k$ converge for every vector
$x=(x_k) \in l^p, 1<p<\infty$ if and only if $b=(b_k) \in l^q, 1/p +1/q=1$ .

This is another standard piece of bookwork. Look it up in any good textbook. The proof uses Hölder's inequality.

Thread: linear functionals, closed sets, convergence

LinkBack

Thread Tools

Display

linear functionals, closed sets, convergence

Similar Math Help Forum Discussions

Convergence of Sequences of Operator and Functionals

Linear Functionals

Linear functionals

Metric Space, closed sets in a closed ball

Continuity & inverse image of closed sets being closed

Search Tags