Weakly Convergent Sequence in Hilbert Space with Convergent Norm is Convergent

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Theorem

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\HH$.

Let $x \in X$ be such that:

$x_n \weakconv x$

and:

$\norm {x_n} \to \norm x$

where $\weakconv$ denotes weak convergence.


Then:

$x_n \to x$


Proof

Let $\norm \cdot$ be the inner product norm for $\struct {\HH, \innerprod \cdot \cdot}$.

From Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence, we have:

$x_n \to x$

if and only if:

$\norm {x_n - x} \to 0$

From Complex Sequence is Null iff Positive Integer Powers of Sequence are Null, it suffices to show that:

$\norm {x_n - x}^2 \to 0$

We have:

\(\ds \norm {x_n - x}^2\) \(=\) \(\ds \innerprod {x_n - x} {x_n - x}\) Definition of Inner Product Norm
\(\ds \) \(=\) \(\ds \innerprod {x_n} {x_n} - \innerprod x {x_n} - \innerprod {x_n} x + \innerprod x x\) Inner Product is Sesquilinear
\(\ds \) \(=\) \(\ds \norm {x_n}^2 - \innerprod x {x_n} - \innerprod {x_n} x + \norm x^2\) Definition of Inner Product Norm

From Weak Convergence in Hilbert Space, we have:

$\innerprod {x_n} x \to \innerprod x x = \norm x^2$

since $x_n$ converges weakly to $x$.

From Weak Convergence in Hilbert Space: Corollary, we also have:

$\innerprod x {x_n} \to \innerprod x x = \norm x^2$

From hypothesis, we have:

$\norm {x_n}^2 \to \norm x$

Then, from Sum Rule for Real Sequences, we have:

$\norm {x_n - x}^2 \to 0$

so:

$x_n \to x$

$\blacksquare$


Sources

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