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I was reading my lecture notes for functional analysis when I came across the following statement:

Let $(e_{n})$ be a total orthonormal sequence in a separable Hilbert space H. The right shift operator, defined as the linear operator $T: H\rightarrow{}H$ such that $Te_{n} = e_{n+1}$ for all n, is bounded.

The statement seems intuitively correct to me, but I find the proof of it quite confusing. The proof goes like this:

Proof: For $\forall x\in{}H$, since $(e_{n})$ is total, write $\displaystyle x=\lim_{n\rightarrow{\infty}}x_{n}$, where $\displaystyle x_{n}=\sum_{k=1}^{n}\lefte_{k}$. Then we have $||Tx_{n}||^{2}=||\sum_{k=1}^{n}\leftTe_{k}||^{2}=||\sum_{k=1}^{n}\lefte_{k+1}||^{2}= \sum_{k=1}^{n}|\left|^{2}$. Therefore $||Tx||^{2}\stackrel{(\ast)}{=}\lim_{n\rightarrow{\infty}}||Tx_{n}||^{2}=\sum_{k=1}^{\infty}|\left|^{2}=||x||^{2}$. Thus, $T$ is bounded and isometric.

However, I think there is something fishy with the proof: In the equality $(\ast)$, I believe the proof is using that $\displaystyle ||Tx||=||T\left(\lim_{n\rightarrow\infty}x_{n}\right)||=||\lim_{n\rightarrow\infty}Tx_{n}||=\lim_{n\rightarrow{\infty}}||Tx_{n}||$. But for the second equality to hold, it is already assuming that T is indeed continuous, which implies boundedness. And that makes it a circular reasoning here...

Is my judgement about the proof right? If this proof is indeed wrong, can anybody suggest a correct way to prove the statement?

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    There is a mistake: $Te_n = e_{n+1}$.2012-04-28
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    @Siminore Oops :)2012-04-28
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    A question: how do you extend the definition of $T$ on the basis to any element of $H$? I mean: what is $Tx$ if $x \in H$?2012-04-28
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    @Siminore Yeah your question got me... It seems without continuity you can't really define $Tx$ for arbitrary $x$... Hmm so are you suggesting that if we ever want to define the right shift operator, we should assume already that it is continuous and bounded?2012-04-28
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    I think this proof is not so meaningless because with assumptions that $T$ is bounded it is also proves that $T$ is isometric.2012-04-28
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    @Siminore Unless the basis we are talking about is a Hamel basis, so that you don't really need to be bothered with the concept of approximation anymore...?2012-04-28
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    Prove first that $T$ is bounded and isometric on the dense subspace of *finite* linear combinations of basis elements. Then there is a unique extension to all of $H$ and this extension will have the same norm and be isometric (why exactly?).2012-04-28
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    @t.b., but you can make extension only for already bounded operators2012-04-28
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    @t.b. Let me guess, Hahn-Banach?2012-04-28
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    @Norbert I think on the subspace of _finite_ linear combination it can be proved to be bounded.2012-04-28
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    Vokram no: Let $D \subset X$ be a dense subset of a metric space and let $Y$ be a complete metric space space. If $f: D \to Y$ is *uniformly continuous* then $f$ has a unique extension to a uniformly continuous function $F: X \to Y$. Uniqueness will enure that the extension of the shift operator is linear. @Norbert: Huh?2012-04-28
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    @t.b. Sorry I overlooked the uniqueness just now, hahn banach doesn't seem to guarantee uniqueness. And moreover the question remains how $Tx$ is defined for $x\notin{}{e_n} $....2012-04-28
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    t.b. Sorry didn't read carefully that you already suggested to prove boundness on dense subspace2012-04-28
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    @t.b. What do you mean by "Uniqueness will enure that the extension of the shift operator is linear."? Sorry I am quite a newbie in functional analysis... And where can I find the proof of your extension theorem? Thanks!2012-04-28
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    enure should have been "ensure". The proof is outlined in Nate's answer, try to do it yourself! See also [here](http://en.wikipedia.org/wiki/Continuous_linear_extension) for a detailed statement of the result and a reference (but any decent book on functional analysis should contain it).2012-04-28

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You are right that there is circularity here.

The problem is in your definition of the right shift operator as "the" linear operator such that $T e_n = e_{n+1}$. In fact, there are many such linear operators. (Using Zorn's lemma, we can extend $\{e_n\}$ to a Hamel basis for $H$ by adding some additional vectors $\{u_\alpha\}$. Then we can define an operator $T$ by setting $T e_n = e_{n+1}$ and setting $T_{u_\alpha}$ to be whatever we want, and this uniquely defines a linear operator.)

So the statement that $T x = \lim T x_n$ will have to be part of the definition of $T$. Following your approach, given $x \in H$, let $x_n = \sum_{k=1}^n \langle x,e_k \rangle e_k$. Then $T x_n$ is unambiguously given by $\sum_{k=1}^n \langle x, e_k \rangle e_{k+1}$. Show that the sequence $\{T x_n\}$ is Cauchy and hence converges to some $y \in H$. Then we can define $Tx$ to be $y$.

Now that $T$ is well defined, one can go ahead and check that $T$ is linear, bounded, and an isometry.

The moral is that defining a linear operator on a total orthonormal set is only well defined if the operator is assumed to be bounded.

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    Thank you so much Nate, it is a enlightening explanation.2012-04-28