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### 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 ...
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### 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....
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...