# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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### What points is the norm Frechet differentiable at

I know the definition of Frechet derivatives - there exists a bounded linear map... Maybe someone could show me a similar example on how to approach questions like these.
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### Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a Banach space for any norm $\|\cdot\|$ on it?

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a complete space for any norm $\|\cdot\|$ on it ?
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### Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
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### Can any uncountable dimensional real vector space be made into a Banach space?

On any real vector space $V$ of uncountable dimension , can we always define a norm such that endowed with that norm , $V$ becomes a complete normed linear space ? ( I know it can be done if $V$ is ...
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### $X$ be a real normed linear space ; if $\mathcal L(X,X)$ is complete then is $X$ also complete?

Let $X$ be a real normed linear space and $\mathcal L(X,X)$ denote the set of all bounded linear operators on $X$ , we know that if $X$ is complete then so is $\mathcal L(X,X)$ ; is the converse true ...
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### Lipschitz map between metric and normed spaces

Let be $F:(X,d)\to V$ a map between $(X,d)$ metric space and $V$ normed space, such that for each $f\in V'$ (linear and continuous), $f\circ F$ is lipschitz map. Show that $F$ is a Lipschitz map. I ...
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### If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} - x_{n_{k-1}}\| < 1/2^k$ [duplicate]

If $X$ is a normed linear space and $(x_n)$ is a Cauchy sequence in $X$, then $x_n$ has a subsequence $x_{n_k}$ which satisfies $$\|x_{n_k} - x_{n_{k-1}}\| < \frac 1 {2^k}$$ for every $k > 0$. ...
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### Let A a subset of X be a finite dimensional linear subspace. Show that A is complete

Let X be a normed space. Let A a subset of X be a finite dimensional linear subspace. Show that A is complete (even if X is not). Using the above show that A is a closed subset of X. For the first ...
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### Equivalence of dual space of normed space X and continuously differentiable functions.

Define that two normed spaces $X$ and $Y$ are equivalent if there exists bounded linear maps $A: X \to Y$ and $B: Y \to X$ such that $A$ and $B$ are inverses of each other. How do you show that there ...
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### Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
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### Dimension and basis of bounded linear maps (products)

Let $X, Y, Z$ be finite dimensional normed spaces with bases $\{x_1,...,x_l\}$, $\{y_1,...,y_m\}$, $\{z_1,...,z_n\}$ respectively. What is the dimension of $\mathcal{L} \{X \times Y; Z\}$ and give ...
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### Show $\sum_{k=1}^\infty|a_k|^q$ converges [duplicate]

Let $1 \leq p < q < \infty$. Show that if $x=(a_k)∈ℓ^p$ i.e. the condition that the series $$\sum_{k=1}^\infty|a_k|^p$$ converges holds, then $x∈ℓ^q$ i.e. $$\sum_{k=1}^\infty|a_k|^q$$ ...
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I am trying to show that the topologies $\tau_p$, $\tau_q$, $\tau_{\infty}$ ($1\leq p<q$) induced by the metrics $$\delta_p(u,v)=\left(\int_0^1 |u-v|^p\right)^{1/p}\quad \delta_q(u,v)=\left(\int_0^... 2answers 89 views ### Closure of a subset of a normed space is equal to the set of limits of sequences in A I'm not sure how to show that the closure of a subset A of a normed space \mathbb{X} is equal to the set of limits of sequences in A. Could someone help me? 1answer 109 views ### Completeness of a finite dimensional linear subspace of X How can I show that a finite dimensional linear subspace F of an arbitrary normed space X is complete, hence closed? 1answer 40 views ### Show that l^p \subseteq l^q for 1 \leq p < q < \infty$$l^p = \{ (a_k)_{k \geq 1} : \sum \limits_{k=1}^{\infty} |a_k|^p < \infty \}$$Since it is said l^p \subseteq l^q, I would have thought we have to show$$\sum \limits_{k=1}^{\infty} |a_k|^q \...
Let $a<b$ be real numbers and $X=C[a,b]$ be the space of continuous functions $f:[a,b] \rightarrow \mathbb R$ Prove that $||f||_1 = \sup _{t \in [a,b]} |f(t)|$ indeed defines a norm on $X$. The ...
There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...