Tagged Questions

2
votes
1answer
76 views

Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?

Let $A$ be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in $A$ such that its spectrum is infinite.
1
vote
0answers
37 views

Baire's theorem [duplicate]

I want to know interesting applications of Baire's Category Theorem. For example existence of no where differentiable function. Can any body tell me some similar applications?
4
votes
2answers
110 views

About Baire's Category Theorem(BCT)

Consider the following theorem known as Baire's Category Theorem (BCT). Theorem.[BCT] A non-empty complete metric space $X$ is not a countable union of nowhere dense sets. I am interested on how to ...
2
votes
1answer
95 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
3
votes
1answer
482 views

Let $X$ be an infinite dimensional Banach space. Prove that every basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
1
vote
2answers
169 views

A question on the proof of Open mapping theorem

I was following the proof of the Open Mapping Theorem in functional analysis in Wikipedia, and I came across a line in the proof that did not make sense. Some notations: $U,V$ are open unit balls in ...
5
votes
1answer
109 views

Is $\ell^1 \subset \ell^2$ meagre? [duplicate]

Possible Duplicate: Prove $\ell_1$ is first category in $\ell_2$ Consider $\ell^2$ with the topology induced by the usual norm. We can easily prove that $\ell^1 \subset \ell^2$. I am ...
4
votes
1answer
120 views

Prove $\ell_1$ is first category in $\ell_2$

Prove that $\ell_1$ is first category in $\ell_2$. I tried to solve this, but had no idea about the approach. Any suggestions are helpful. Thanks in advance.
53
votes
18answers
3k views

Your favourite application of the Baire category theorem

I think I remember reading somewhere that the Baire category theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
85
votes
1answer
2k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...