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Math Help - Isomorphism beetwenn vector space and sub space

  1. #1
    Junior Member
    Joined
    Sep 2008
    From
    Freiburg i. Brgs. Germany
    Posts
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    Question Isomorphism beetwenn vector space and sub space

    Hi,

    I have to find a vector space V with a real sub space U and a bijective linear map.

    Here my Ideas and my questions:

    If the linear map is bijective, than dim V = dim U

    Because U is a real sub space the only way to valid this constraint is if the dimension is infinity. I wrote:

    U \subseteq V ~ f: U \rightarrow V bijective

    dim ~ U = dim ~ V = \infty

    U = x_{1}e_{1} + x_{2}e_{2} + x_{i}e_{n} = \sum\limits_{i,n=1}^{\infty} x_{i}e_{n} \ x_{i} \in k, ~ e_{n} \in U, ~ i,n \in \mathbb{N}

    V = x_{1}e_{1} + x_{2}e_{2} + x_{j}e_{m} = \sum\limits_{j,m=1}^{\infty} x_{j}e_{m} \ x_{j} \in k, ~ e_{m} \in U, ~ j,m \in \mathbb{N}

    \sum\limits_{i,n=1}^{\infty} x_{i}e_{n}   = \sum\limits_{j,m=1}^{\infty} x_{j}e_{m} ~ \Leftrightarrow ~ f: U \ \rightarrow V ~ isomorphism

    1.) Are my minds up to now correct?

    2.) How to go on? Maybe a complete induction? But I have different indices.

    Thank you
    all the best
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  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi.

    I think I understand what you mean, but take care, the sums \sum\limits_{i,n=1}^{\infty}x_{i}e_{n}\ \ \& \sum\limits_{j,m=1}^{\infty}x_{j}e_{m} are exactly the same, changing indices' letters doesn't change anything.

    But indeed if you want to have a vector space in bijection with one of his proper subspaces, then they must have the same infinite dimension.

    Why don't you take an example, like \mathbb{R}[X], whose dimension in infinite (countable), and its subspace \{P(X) \in \mathbb{R}[X]\ ;\ \deg (P(X)) \in 2\mathbb{N} \}.

    Is there an isomorphism between them?
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