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Let X be a normed $\mathbb{R}$-space, $\gamma>0$,$(x_i)_{i\in \mathbb{N}}$ a sequence in X and $(a_i)_{i\in \mathbb{N}}$ a sequence in $\mathbb{R}$. Show the equality of the following two statements:

i) There is a $F\in X'$ with $||F||_{X'}\le \gamma$ and $F(x_i)=a_i \forall i\in \mathbb{N}$

ii) $\forall n\in \mathbb{N}$ and all sequences $(\beta_i)_{i\in\mathbb{N}}$ in $\mathbb{R}$: $$|\sum_{i=1}^n\beta_ia_i|\le\gamma||\sum_{i=1}^n\beta_ix_i||_X$$

As a part of this exercise I need to show the following:

$$|Fx|\le C||x||_X \forall x\in X$$

$$\Leftrightarrow|\sum_{i=1}^n\beta_iFx_i|\le C||\sum_{i=1}^n\beta_ix_i||_X$$

My problem here is the norm, which includes the sum, so I can't estimate upwards using "$|Fx|\le C||x||_X \forall x\in X$". Once I've shown this inequality I think I've got the rest of the exercise. I wanted to show $i) \Rightarrow ii)$ using the continuity of F and $ii) \Rightarrow i)$ using Hahn-Banach, but I can't do either without the above.

Can someone help me with this final step?

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Tobi92sr
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1 Answers1

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You don't have an equivalence

$$(\forall x\in X)\bigl(\lvert Fx\rvert \leqslant C \lVert x\rVert_X\bigr) \iff \biggl\lvert \sum_{i=1}^n\beta_iFx_i\biggr\rvert\leqslant C \biggl\lVert \sum_{i=1}^n\beta_ix_i\biggr\rVert_X$$

in general because the right hand side only tells you something about the subspace $Y := \operatorname{span} \{ x_i : i \in \mathbb{N}\}$. The implication from the left to the right however always holds, by inserting $ x = \sum_{i = 1}^n \beta_i x_i$.

But the right hand side tells you that the restriction of $F$ to $Y$ is a continuous linear functional of norm $\leqslant C$.

So, for the direction $ii) \implies i)$, define $F_0$ on $Y$ by setting

$$F_0\biggl(\sum_{i = 1}^n \beta_i x_i\biggr) = \sum_{i = 1}^n \beta_i a_i.$$

Use the condition $ii)$ to show that $F_0$ is well-defined. Then it follows easily that $F_0 \colon Y \to \mathbb{R}$ is a continuous linear functional of norm at most $\gamma$. Then use Hahn-Banach to extend $F_0$ to all of $X$.

Daniel Fischer
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  • How does $ii)$ imply that $F_0$ is well-defined? – Tobi92sr Nov 10 '17 at 01:03
  • If you have two ways to write $x$ as a linear combination of the $x_i$, say $$\sum_{i = 1}^n \beta_i x_i = x = \sum_{j = 1}^m \gamma_j x_j,$$ you must see that $$\sum_{i = 1}^n \beta_i a_i = \sum_{j = 1}^m \gamma_j a_j.$$ That is, extending the shorter coefficient sequence by $0$, $$\sum_{k = 1}^L (\beta_k - \gamma_k)a_k = 0$$ where $L = \max {m,n}$. Can you see how that is implied by ii)? – Daniel Fischer Nov 10 '17 at 11:39