Thread: Analysis Proof - Convergence and scalar product

1. Analysis Proof - Convergence and scalar product

Problem:
Let ${ u_{k} }$ be a sequence in $\mathbb{R}^n$ and let $u \in \mathbb{R}^n$. Suppose that for every $v \in \mathbb{R}^n$,

$\lim_{k \to \infty} \left\langle u_{k}, v \right\rangle = \left\langle u, v \right\rangle$

Prove that ${ u_{k}}$ converges to u.

========================
Let $p_{i}$ be the ith coordinate function $\mathbb{R}^n \rightarrow \mathbb{R}$ and let $u, v \in R^n$

For every index i with $1 \leq i \leq n$ let $p_{i}(u) = \left\langle u, e_{i} \right\rangle$ where $e_{i} \in R^{n}$ and which the ith component is 1 and all other components are 0.

Since we are given that

$\lim_{k \to \infty} \left\langle u_{k}, v \right\rangle = \left\langle u, v \right\rangle$

$\implies \left\langle u , v \right\rangle = \sum_{i}^n {p_{i}(u)p_{i}(v)}$

From here, I do not know how to show that $\lim_{x \to \infty} u_{k} = u$

Thank you for reading. Any help is greatly appreciated.

2. Originally Posted by Paperwings
Problem:
Let ${ u_{k} }$ be a sequence in $\mathbb{R}^n$ and let $u \in \mathbb{R}^n$. Suppose that for every $v \in \mathbb{R}^n$,

$\lim_{k \to \infty} \left\langle u_{k}, v \right\rangle = \left\langle u, v \right\rangle$

Prove that ${ u_{k}}$ converges to u.

========================
Let $p_{i}$ be the ith coordinate function $\mathbb{R}^n \rightarrow \mathbb{R}$ and let $u, v \in R^n$

For every index i with $1 \leq i \leq n$ let $p_{i}(u) = \left\langle u, e_{i} \right\rangle$ where $e_{i} \in R^{n}$ and which the ith component is 1 and all other components are 0.

Since we are given that

$\lim_{k \to \infty} \left\langle u_{k}, v \right\rangle = \left\langle u, v \right\rangle$

$\implies \left\langle u , v \right\rangle = \sum_{i}^n {p_{i}(u)p_{i}(v)}$

From here, I do not know how to show that $\lim_{x \to \infty} u_{k} = u$

Thank you for reading. Any help is greatly appreciated.
Pick $\bold{v} = \bold{e}_k$ where $\{ \bold{e}_1,...,\bold{e}_n\}$ is the standard basis.
Now write $\bold{u}=(a_1,...,a_n)$ then $\left< \bold{u},\bold{e}_k\right> = a_k$.
This means the sequence of $\{ \bold{u}_j\}$ at the $k$ coordinate converges to $a_k$.
Therefore the limit of $\{ \bold{u}_j\}$ is $(a_1,...,a_k)$.

3. Originally Posted by Paperwings
Problem:
Let ${ u_{k} }$ be a sequence in $\mathbb{R}^n$ and let $u \in \mathbb{R}^n$. Suppose that for every $v \in \mathbb{R}^n$,

$\lim_{k \to \infty} \left\langle u_{k}, v \right\rangle = \left\langle u, v \right\rangle$

Prove that ${ u_{k}}$ converges to u.

========================
Let $p_{i}$ be the ith coordinate function $\mathbb{R}^n \rightarrow \mathbb{R}$ and let $u, v \in R^n$

For every index i with $1 \leq i \leq n$ let $p_{i}(u) = \left\langle u, e_{i} \right\rangle$ where $e_{i} \in R^{n}$ and which the ith component is 1 and all other components are 0.

Since we are given that

$\lim_{k \to \infty} \left\langle u_{k}, v \right\rangle = \left\langle u, v \right\rangle$

$\implies \left\langle u , v \right\rangle = \sum_{i}^n {p_{i}(u)p_{i}(v)}$

From here, I do not know how to show that $\lim_{x \to \infty} u_{k} = u$

Thank you for reading. Any help is greatly appreciated.
For fixed $i$ what is $\lim_{k \to \infty} p_i(u_k)$ ? So what is $\lim_{k \to \infty} p_i(u_k)p_i(v)$ ? And:

$\lim_{k \to \infty} \sum_{i=1}^n p_i(u_k)p_i(v)=\lim_{k \to \infty} \langle u_k,v \rangle$ ??

RonL

4. To CaptainBlack, For $\lim_{k \to \infty} p_i(u_k)$, then $\lim_{k \to \infty} p_i(u_k) = p_{i}(u)$.

As a result, then $\implies \lim_{k \to \infty} p_i(u_k)p_i(v) = p_i(u) p_i(v) \neq \left\langle u, v \right\rangle$

5. Originally Posted by Paperwings
To CaptainBlack, For $\lim_{k \to \infty} p_i(u_k)$, then $\lim_{k \to \infty} p_i(u_k) = p_{i}(u)$.

As a result, then $\implies \lim_{k \to \infty} p_i(u_k)p_i(v) = p_i(u) p_i(v) \neq \left\langle u, v \right\rangle$
$\left\langle u, v \right\rangle=\sum_i p_i(u)p_i(v)=\sum_i \lim_{k \to \infty} p_i(u_k)p_i(v)=\lim_{k \to \infty} \sum_i p_i(u_k)p_i(v)=\lim_{k \to \infty} \left\langle u_k, v \right\rangle$

The sum and limit can be interchanged because the sum is a finite sum.

RonL