# Linear forms and complete metric space

by quasar987
Tags: forms, linear, metric, solved, space
 Sci Advisor HW Helper PF Gold P: 4,771 1. The problem statement, all variables and given/known data Question: Let L be a linear functional/form on a real Banach space X and let {x_k} be a sequence of vectors such that L(x_k) converges. Can I conclude that {x_k} has a limit in X? It would help me greatly in solving a certain problem if I knew the answer to that question. 3. The attempt at a solution The natural approach is to try to show that {x_k} is Cauchy. Since the sequence of real numbers {L(x_k)} converges, then it is Cauchy, so for n,k large enough, $$|L(x_k)-L(x_n)|=|L(x_k - x_n)|<\epsilon$$ Now what??
 P: 371 If L is not invertible what can happen?
 Sci Advisor HW Helper P: 2,020 edit: removing my too explicit hint.
 P: 371 Linear forms and complete metric space Boo. I'm sure he could have figured it out on his own. What is the particular quandary? Edit: Suppose L is invertible! What can you say then?
 Sci Advisor HW Helper PF Gold P: 4,771 Too late morphism! :D
 Sci Advisor HW Helper PF Gold P: 4,771 Anyway, guys, in the problem I'm working on, I must show that in a particular situation, the x_k do have a limit in X. Are you willing to help with the general problem? If so, I will type it out. Not very long, but a little complicated notation-wise.
 Sci Advisor HW Helper PF Gold P: 4,771 It's at the core, a problem on measure theory. Let $$\mathcal{L}^2(\mathbb{R})$$ denote the space of square-integrable functions f on R (with respect to the Lebesgue sigma-algebra and Lebesgue measure $\lambda$) and let $$\mathcal{L}^2_{\lambda}(\mathbb{R})$$ denote the space of their equivalence classes f where f=g if f=g almost everywhere. [f]+[g]=[f+g] and c[f]=[cf] are well defined. Now with the norm $$||\mathbf{f}||_2=\int_{-\infty}^{+\infty}|f(x)|^2dx$$ $$(\mathcal{L}^2_{\lambda}(\mathbb{R}),||||_2)$$ is a real Banach space. A continuous linear form on $$(\mathcal{L}^2_{\lambda}(\mathbb{R}),||||_2)$$ is a linear form $$L:\mathcal{L}^2_{\lambda}(\mathbb{R})\rightarrow \mathbb{R}$$ such that $$||L||=\sup\left\{\frac{|L(\mathbf{f})|}{||\mathbf{f}||_2}: \mathbf{f}\neq \mathbf{0}\right\}=\sup\left\{|L(\mathbf{f})|:||\mathbf{f}||_2=1\right\ }<+\infty$$ Now consider a sequence $$\mathbf{g}_k\in \mathcal{L}^2_{\lambda}(\mathbb{R})$$ be such that $$||\mathbf{g}_k||_2=1$$ and $$\lim_{k\rightarrow\infty}L(\mathbf{g}_k)=||L||$$ Show that $$\mathbf{g}_k$$ has a limit in $$\mathcal{L}^2_{\lambda}(\mathbb{R})$$.
 Sci Advisor HW Helper P: 2,020 Is L a fixed functional? If so, I don't think this is true. Try constructing a counterexample using the zero functional.
 Sci Advisor HW Helper PF Gold P: 4,771 Apologies! There is an additional hypothese! L is a non-identically vanishing functional!
 Sci Advisor HW Helper P: 2,020 I'm tempted to use the Riesz representation theorem: we know there is a nonzero f in L^2 such that L(g)= for all g in L^2, and ||L||=||f||_2. Now consider ||f - g_n||$_2^2$. (Hint: apply the polarization identity, and use the fact that -> ||f||.) Try to see if you can guess what (g_n) converges to using this.
 Sci Advisor HW Helper PF Gold P: 4,771 We are not to use this theorem in this problem, because in a sense, the whole problem sheet comes down to showing explicitely that the Riesz representation theorem hold in the case of L². No prior knowledge of functional analysis should be needed to do this problem. Someone told me he succeeded in answering this question by effectively proving that the sequence g_k was Cauchy!
 Sci Advisor HW Helper P: 2,020 Did you manage to do it without Riesz?
 Sci Advisor HW Helper PF Gold P: 4,771 Yes, with the help of aforementioned person. :) With the parallelogram identity, we reduce the problem to showing $$||\mathbf{g}_k+\mathbf{g}_{k+p}||_2^2 \rightarrow 4$$ Then notice that because ||L|| is the sup, $$\frac{L(\mathbf{g}_k+\mathbf{g}_{k+p})}{||\mathbf{g}_k+\mathbf{g}_{k+p} ||_2}\leq ||L|| \ \ \ \ \ \ (*)$$ On the other hand, write out the facts that L(g_k)-->||L|| and L(g_{k+p})-->||L|| and add the inequalities to obtain $$L(\mathbf{g}_k+\mathbf{g}_{k+p})>2(||L||-\epsilon')$$ Combine with equation (*) to obtain an inequality involving $$||\mathbf{g}_k+\mathbf{g}_{k+p}||_2$$ and $$\epsilon'$$. Show that to any $$\epsilon>0$$, you can find an $$\epsilon'(\epsilon)$$ such that $$4-||\mathbf{g}_k+\mathbf{g}_{k+p}||_2^2<\epsilon^2$$ for k large enough.

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