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 ?

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- May 8th 2010, 05:09 PMneset44Proof for banach Space 2
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 ? - May 8th 2010, 08:15 PMDrexel28
- May 9th 2010, 01:19 AMneset44
sequence in banach space

- May 9th 2010, 02:11 AMOpalg
What Drexel28 is asking is: What does the dot mean in ? 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.

- May 9th 2010, 02:54 AMHallsofIvy
In fact, although you titled this "proof for Banach space 2", you only said "In a normed linear space Xn". Not all normed spaces

**are**Banach spaces. Please state the problem in full. - May 10th 2010, 02:37 AMneset44
Xn and Yn are in Banach space. So in a normed linear space Xn converges to X and Yn converges to Y then prove that XnYn (multiply) converges to XY.

That is the full problem. - May 10th 2010, 02:44 AMOpalg
There is no operation of multiplication defined in a Banach space, so this question makes no sense.

In order for a product 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. - May 10th 2010, 02:59 AMneset44
i changed product with standart multiply as you see. but anyway;

This is the full problem .

It says on theorem :

In a normed linear space Xn coverges to X and Yn converges to Y => XnYn converges to XY . Prove it ?

Nothing more - May 10th 2010, 04:14 AMHallsofIvy
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". - May 10th 2010, 04:21 AMneset44
i asked and My teacher said . It is a scalar multiplication

- May 10th 2010, 04:25 AMHallsofIvy
Are you saying that {Xn} is a sequence of scalars that converge to X and {Yn} is a sequence of vectors in a Banach space that converge to Y? That was

**not**what you said initially! - May 10th 2010, 01:06 PMneset44
as a result that what he given to me Xn converges to x and Yn converges to y then XnYn converges to xy ?

there is not any more information. then he said between Xn and Yn multiplication is scalar ...

that is :S(Angry) - May 10th 2010, 08:22 PMJose27
- May 11th 2010, 01:17 AMmabruka
My guess is that he is talking about a Banach Algebra.