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Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

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Given a point $x$ and a closed subspace $Y$ of a normed space, must the distance from $x$ to $Y$ be achieved by some $y\in Y$?

I think no. And I am looking for examples. I would like a sequence $y_n$ in $Y$ such that $||y_n-x||\rightarrow d(x,Y)$ while $y_n$ do not converge. Can anyone give a proof or an counterexample to this question?
Spook
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Example to prove that $ C^1[0,1] $ is not a Banach space for the uniform norm?

The space $ C^1[0,1] $- the space of all continuously differentiable functions on $ [0,1]$ is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $ since the uniform limit of a continuously differentiable function need not be…
ccc
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there exists a sequence $x_n$ such that $\| x_n \|=1$ and $x_n$ converges weakly to $0$.

Let $X$ be a reflexive Banach space of infinite dimension. a) Prove that there exists a sequence $x_n$ such that $\| x_n \|=1$ and $x_n$ converges weakly to $0$. b) Let $x_n$ be a sequence such that $\forall f \in X' \quad \exists…
user62138
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Prove that if a linear operator is continuous, then it is bounded.

I'm trying to prove that if a linear operator is continuous, then it is bounded. Let $T:V\to W$. Let us assume it is continuous. Then for any $\epsilon>0$, $\|T(x-x_0)\|<\epsilon$ if $\|x-x_0\|<\delta$ for some $\delta\in \Bbb{R}$. If $T$ is…
user67803
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Open linear subspace of a Hilbert space.

Does there exist any open linear (vector) subspace of a Hilbert space? I could not think of any example. Actually, I was reading the book by Simmons, there almost in every theorem it assumed that "If M is a closed linear subspace".It seemed natural…
epsilon_delta
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Two exercises in functional analysis.

I have met two questions which, after some attempts have I have yet been able to solve nor find any available solutions online. Could anyone please offer me some insights? Let $ H $ be a Hilbert space over $\mathbb{C}$ and $T \in B(H,H)$ an unitary…
Meagain
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Exercise: Application of Hahn-Banach Theorem

I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it! Exercise: Let $\{x_1,\dots,x_n\}$ be a set of linearly independent elements of a normed vector space $X$. Let $c_1,\dots,c_n \in \mathbb{C}$. Show…
DoubleTrouble
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Every finite-dimension subspace of $\mathcal{X}$ is closed.

Background Information: (Folland)Theorem 5.8 - Let $\mathcal{X}$ be a normed vector space. a.) If $M$ is a closed subspace of $\mathcal{X}$ and $x\in \mathcal{X}\setminus M$, there exists $f\in\mathcal{X}^*$ such that $f(x)\neq 0$ and $f(M) =…
Wolfy
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Is this an inner product on $L^1$?

I know that $\int f(x) \overline{g(x)} dx$ is an inner product on $L^2$. But is it one on $L^1$? I think it isn't, but I am have had difficulty figuring out which defining property is violated. Thanks in advance for any pointers!
Not Rudin
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Are the polynomial functions on $S^1$ dense in $C(S^1,ℂ)$?

A friend of mine came up with this problem: Let $S^1$ be the unit circle in $ℂ$ and $P$ the space of polynomial functions $S^1 → ℂ$ (with complex coefficients). Is $P$ dense in $C(S^1,ℂ)$? Stone–Weierstraß is not applicable because $P$ is not closed…
k.stm
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projection theorem for Banach spaces

In a remark to the projection theorem for Hilbert spaces I read this conjecture of a more general projection theorem: Let $X$ be a reflexive Banach space and $K\subset X$ nonempty, closed and convex. Then for every $x\in X$ there exists $y\in K$…
dinosaur
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$a: E\times F\to G$ bilinear separately continuous implies continuous?

Let $E$, $F$ and $G$ be Banach spaces and let $a$: $E \times F \to G$ be a bilinear map which is separately continuous, that is $$\forall x \in E \textrm{ the map } y \mapsto a(x,y) \textrm{ is continuous}$$ and $$\forall y \in F \textrm{ the map }…
JOHN T MENDY
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Why are inner products in RKHS linear evaluation functionals?

I'd like to know why inner products in Reproducing kernel Hilbert spaces are (linear) evaluation functionals. I understand that inner products are linear functionals, and I know what an evaluation functional is, I just can't explain why an inner…
Olumide
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Continuity in weak convergence implies continuity in norm convergence

Let $X$ and $Y$ be norm spaces, and let $T:X\to Y$ be a linear transformation which is continuous under weak convergence. That is, if $\forall x^\star\in X^\star:x^\star x_n\to x^\star x $ then $\forall y^\star \in Y^\star:y^\star Tx_n \to y^\star…
Shai Deshe
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Why unit open ball is open in norm topology, but not open in weak topology?

Why unit open ball is open in norm topology, but not open in weak topology? I will be grateful for any explanation. Edit: Obviously in infnite dimensional spaces.
luka5z
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