Linked Questions

0
votes
1answer
184 views

Are normed vector spaces with countably infinite algebraic dimension never complete? [duplicate]

I'm not quite sure about this but I wrote down during a lecture that a normed vector space with an algebraic basis containing a countably infinite number of elements is never complete. What I mean by ...
1
vote
0answers
99 views

Proving Banach space is finite in Baire space [duplicate]

Let $B$ be a Banach space where the dimension of the underlying vector space is countable. using the Baire category Theorem, prove that the dimension of the underlying vector space is, in fact, finite....
0
votes
0answers
89 views

Basis for the linear spacer $\ell^p$ [duplicate]

Is there any well known basis (Hamel basis) for the vector space $\ell^p$? And what about the cardinality of such basis? Is it countable?
1
vote
0answers
51 views

Question about Hamel's basis. [duplicate]

I'd like to prove that if I consider $X$ Banach space with $\dim X =\infty$, then $X$ can't have a countable Hamel's basis. Someone can give me a hint? Even a counterexample is enough. I think this ...
189
votes
25answers
24k 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 ...
64
votes
5answers
15k views

What is the difference between a Hamel basis and a Schauder basis?

Let $V$ be a vector space with infinite dimensions. A Hamel basis for $V$ is an ordered set of linearly independent vectors $\{ v_i \ | \ i \in I\}$ such that any $v \in V$ can be expressed as a ...
34
votes
3answers
10k views

Every proper subspace of a normed vector space has empty interior

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric $||...
25
votes
2answers
3k views

Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
9
votes
3answers
819 views

Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
11
votes
1answer
8k views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a finite-...
1
vote
2answers
1k views

Two problems: When a countinuous bijection is a homeomorphism? Possible cardinalities of Hamel bases? [closed]

Let $X$ and $Y$ be topological spaces and let $f : X\rightarrow Y$ be a continuous bijection. Under which of the following conditions will $f$ be a homeomorphism? (a) $X$ and $Y$ are complete metric ...
2
votes
3answers
1k views

Is the cardinality of the basis of a vector space $V$ having infinite dimension necessarily countable infinite?

Suppose a vector space $V$ has infinite dimension, $\dim V = \infty$. Is the cardinality of the basis of $V$ neccesarily countable infinite? If a vector space has infinite dimension does $V$ then ...
1
vote
1answer
1k views

Range of any projection is closed.

Let $X$ be a Banach space and $P$ a projection. Show that the range of any projection is a closed subspace. Can I use the fact that a Banach space is complete and thus closed and that $P = P^2$ to ...
2
votes
3answers
737 views

Prove that $(C([0,1]),\lVert \cdot \rVert_\infty)$ is infinite dimensional

Let $C([0,1])$ equipped with $\lVert \cdot \rVert_\infty$ be set of all functions continuous on $[0,1]$. Prove that $C([0,1])$ is not finite dimensional. There is a theorem which states that a normed ...
1
vote
1answer
766 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...

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