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Let $H_1, H_2$ be Hilbert spaces and $(x,y) \in H_1 \times H_2$ whith $x \neq 0$.

  1. It exists $f \in H_1': \lVert f \rVert = 1 \land f(x) = \lVert x \rVert$
  2. Construct $T: H_1 \rightarrow H_2$ such that $Tx = y$

I am trying to prove this excercise from my Functional analysis book. For the first, I thought that it must be a corolary of Hahn-Banach but since we do not have subspaces I do the following: take $g_x \in H_1: g_x(z) = \langle z, x \rangle$ . For all $z \in H_1: g_x ( z - \frac{g_x(z)}{g_x(x)}x) = 0 \implies H_1 = \Bbb{K}x + \ker g_x$ . Define $f: H_1 \rightarrow \Bbb{K}, f(\lambda x + v) = \lambda \lVert x \rVert$ . It is easy to see that it is linear and that for all $(\lambda , v) \in (\Bbb{K}\setminus \{0\}) \times \ker g_x : |f(\lambda x + v)|^2 = |\lambda|^2 \cdot \lVert x \rVert^2 = \lVert \lambda x \rVert^2 = \langle \lambda x, \lambda x \rangle = \langle \lambda x + v , \lambda x + v\rangle - \langle v , v \rangle = \lVert \lambda x + v\rVert^2 - \lVert v \rVert^2 \leq \lVert \lambda x + v\rVert^2 \implies |f(\lambda x + v)| \leq \lVert \lambda x + v \rVert \implies \lVert f \rVert \leq 1$

Trivially we have $\lVert x \rVert = |f(x)| \leq \lVert f \rVert \cdot \lVert x \rVert \implies 1 \leq \lVert f \rVert$, so $\lVert f \rVert = 1$ .

The problem is that I do not know how to proceed for 2, I have the intuition that might be related to adjoint operators theory but I do not have idea on which theorems use, so any possible help would be appreciated.

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1 Answer 1

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Concerning $1.$ the linear functional $f_x(z)=\langle z,{x\over \|x\|}\rangle$ satisfies the requirements.

Concerning $2.$ take $Tz={1\over \|x\|}f_x(z)y.$

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  • $\begingroup$ Thanks for your answer! It is really helpful $\endgroup$ Commented May 10 at 19:13
  • $\begingroup$ You welcome. The conclusion holds for normed spaces $X\times Y.$ This time for a fixed pair $(x,y)$ with $x\neq 0$ by the Hahn-Banach theorem there is $f_x$ satisfying the requirements. Then $T$ is defined in the same way as for Hilbert spaces. $\endgroup$ Commented May 10 at 19:19

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