# Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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### Weak vs. Weak* Convergence of Bounded Nets

Let $X,Y$ be Banach spaces. In V. Paulsen's book on completely positive maps, he shows that $B(X,Y^\ast)$ is a dual space as follows: For $x \in X, y \in Y$ let $x \otimes y \in B(X, Y^\ast)^\ast$ be ...
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### Show that the vector space $l_1$ with the norm defined by $\| (a_1,a_2,..)\|= \sum_{k=1}^{\infty}|a_k|$ is a Banach space. [duplicate]

Show that the vector space $l_1$ with the norm defined by $\| (a_1,a_2,..)\|= \sum_{k=1}^{\infty}|a_k|\leq\infty$ is a Banach space. My work- Let $(a_{i})=(a_{{i}{1}},a_{{i}{2}},a_{{i}{3}}...)$ be a ...
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### Compute norm in a Banach space

Let $I = [0, 1] \subset \mathbb{R}$ and the scalar field is $\mathbb{R}$. For a Banach space $C(I)$, let $\Lambda(f)=\int_{0}^{1}\left(9 t^{4}-18 t^{3}+11 t^{2}-2 t\right) f(t) d t$ I would like to ...
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### Proving every linear operator in the dual space is compact.

How can I prove that for $X$ being a Banach Space, every $\phi \in X^*$ is compact?
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### Supersets of unit ball in infite dimension

Let $X$ be a Banach space with infinite dimension. Is it true that every superset of the unit ball $B_X(0)$ in $X$ is not compact. In finite dimensions this is not true. I couldn't find any ...
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### Why is the space of linear operators to a Banach Space complete

I know this has been answered before, but I wanted to understand a particular proof. We have a space X and a Banach Space Y. We take a sequence of bounded linear operators from X to Y i.e. $T_n$ which ...
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### $Y$ is Banach; if $X$ is a topological space, $C_b(X,Y)$ is Banach [closed]

Let $X$ is a topological space and $Y$ is normed space. $C_b(X,Y)$ is Banach iff $Y$ is Banach.
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### Unique extension of a bounded linear operator : Reference request

Does someone know a textbook which states the theorem considered in this question? Preferably, such a book should be released recently rather than many years ago as I'd prefer a source which does not ...
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### Homogeneous analytic function on Banach space

Definition: Let $X$ be a complex Banach space. We say $f: X\rightarrow \mathbb{C}$ is holomorphic if it satisfies the following: 1.$f$ is locally bounded. That means given $x\in X$, there exists an ...
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### Example of function that is Gâteaux-differentiable but not Fréchet-differentiable

I am looking for an example of a function that is Gateaux-differentiable but not Fréchet-differentiable. I know that there is a lot of example of function $f: \mathbb R^2 \to \mathbb R$ that satisfies ...
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### Norm of Linear operator as a limit

It is known that a linear operator on Banach space not necessarily attains its norm (meaning that there is no element $x$ s.t. $\|Tx\| = \|T\|$). I would like to clarify that there is a sequence that ...
### Why is $C^{k}([a,b],||\cdot||)_{C^{k}}$ a Banach space? [closed]
If $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ and $||\cdot||_{C^{k}}=\sum_{i=0}^{k}||f^{(k)}||_{\infty}$, why is this a complete norm on $C^{k}([a,b]\longrightarrow\mathbb{K})$?
Let $T$ be a linear operator between Banach spaces $\mathscr{X}$ and $\mathscr{Y}$ which is defined everywhere in $\mathscr{X}$. Could $T$ have a closed null space $N(T)=\{x \in \mathscr{X}|Tx=0\}$ ...