Prove that, In a normed linear space Xn converges to X and Yn converges to Y then XnYn converges to XY .
Xn and Yn are sequence in banach space.
Anyone idea ?
What Drexel28 is asking is: What does the dot mean in $\displaystyle X_n.Y_n$? It looks as though it is supposed to be an inner product. But a Banach space doesn't have an inner product (unless it is a Hilbert space), so it's hard to know what this problem is supposed to be about.
There is no operation of multiplication defined in a Banach space, so this question makes no sense.
In order for a product $\displaystyle X_n.Y_n$ to be defined, there must be some additional structure in the space. It has to be either a Hilbert space (with an inner product), or a Banach algebra (with a multiplicative structure).
Also, as HallsofIvy has pointed out, a Banach space is not the same as a normed linear space.
One more time and then we give up!
A "Banach space" is a vector space having a norm such that "Cauchy sequences" converge.
While "scalar multiplication" and "vector addition" are defined in any vector space, multiplication of vectors is NOT generally defined. There is no "standard multiply"! It is impossible to answer your question with knowing how you are defining "XnYn".
You really need to improve you english writing skills.
That aside, if you meant $\displaystyle (x_n) \subset \mathbb{C}$ (or $\displaystyle \mathbb{R}$ it works the same) and $\displaystyle (y_n) \subset X$ (where this last one is a normed space over whichever field you pick from the above) such that $\displaystyle |x_n - x| \rightarrow 0$ and $\displaystyle \| y_n - y \| \rightarrow 0$ then
$\displaystyle \| x_ny_n - xy \| \leq \| x_ny_n -x_ny\| + \|x_ny-xy\| \leq M\| y_n-y\| + |x_n-x| \|y\| \rightarrow 0$
where $\displaystyle |x_n|\leq M$ for all $\displaystyle n$.