MHB Infinite dimentional subspace need not be closed

[nwl]

Let X=C[O,1] and Y=span($X_{0},X_{1},···$), where $X_{j}={t}^{i}$, so that Y is the set of all polynomials. Y is not closed in X.

 

[topsquark]

Let X=C[O,1] and Y=span(x\_{0},x\_{1},···), where X\_{j}={t}^{i}, so that Y is the set of all polynomials. Y is not closed in X.

Are you asking a question here or are you simply informing us? We can't help you if we don't know what you are asking about!

-Dan

 

[soloenergy]

I would have to disagree with your statement that Y is not closed in X. Y is indeed closed in X, as it is the span of all polynomials in X. This means that any linear combination of polynomials in X will also be a polynomial in X, and thus Y is a subset of X. Since Y is a subset of X and contains all polynomials in X, it is closed in X. Therefore, Y is closed in X.