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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 .

52582 questions
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Is this space Dense in $ C[0,1]$

Is the space $\{f\in C[0,1]\mid \int_0^1f\neq 0\}$ dense in $C[0,1]$ with sup-norm topology. I think yes, because it is the inverse image of the set $\mathbb{R} \setminus \{0\}$
vidyarthi
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Help showing subadditivity of a map

I'm stuck with the following problem. Show that the map: $$ r(x)=\inf\limits_{k\in\mathbb{N}}\limsup\limits_{m\to\infty}\frac{1}{k}\sum\limits_{j=0}^{k-1}S^j(x)(m) $$ is subadditive on $\ell_\infty(\mathbb{N})$. Here…
Tim
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A continuous bijection from $l_2 $ onto a subset of $l_2$ whose inverse is everywhere discontinuous.

I was reading an article from AMM, titled, A continuous bijection from $l_2 $ onto a subset of $l_2$ whose inverse is everywhere discontinuous. In this he constructed the function $T:l_2\rightarrow l_1$ as $T(x)=$$(\sigma(x_1)x_1^2,…
User
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The space is complete iff the subspace and the quotient space is complete.

Let $F$ be a closed subspace of a normed space. How to prove that $E$ is complete iff $F$ and $E/F$ are both complete ?
learning_math
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Operators with compact resolvent

This should be a basic, or even stupid, question, but I am really confused, and I cannot find any webpage that addresses my question. From wikipedia (https://en.wikipedia.org/wiki/Resolvent_formalism), an operator $A$ has compact resolvent iff $(A -…
Alex
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Application of Hahn-Banach Theorem

Let $E$ be Banach space over $\mathbb{R}$. Let $u$ and $v$ be such that $||u||=||v||=1$ and $||2u+v||=||u-2v||=3$. How do we show that there exists a linear functional $f$ defined on all of $E$ such that $||f||=1$, $f(u)=1$ and $f(v)=1$?
VJunior
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Question about proof of Hahn-Banach lemma

I think they do something unnecessary in my notes in the proof of the following lemma: The idea of the proof is to partially order the set $\Sigma$ of pairs $(X_i, f_i)$ where $X_i$ is a linear subspace of $X$ containing $Y$ and $f_i : X_i \to…
Rudy the Reindeer
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Application of Hölder's inequality on $\mathbb R^n$

Let $1 \leq p < q \leq \infty$ ($p$ and $q$ are not otherwise related). Given $\|x\|_q\leq\|x\|_p $ $\forall$ $ x \in \mathbb R^n$ how can I use Hölder's inequality to show $\|x\|_p\leq n^{\frac{1}{p}-\frac{1}{q}}\|x\|_q$ . I can see this link is a…
havakok
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Understanding the proof: a linear functional is continuous if and only if it is bounded.

Let $X$ be a normed space. Prove that a linear functional $f:X \to \mathbb{R}$ is continuous if and only if there is a number $ c \in {0, \infty}$ such that $$|f(x)| \leq c\|x\|$$ for all $x \in X$ This is the proof: Suppose there is a number $ c…
Al jabra
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Cauchy-Schwarz inequality for dual pairing?

Suppose we have a Hilbert space $X$ and its dual $X^*$. Given a dual pairing $$_{X^*}\langle x,y\rangle_X,$$ does there exist a sort of Cauchy-Schwarz inequality so that $|\langle x, y\rangle|\leq ||x||_{X^*}||y||_{X}$?
user23793
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When are two norms equivalent on a Banach space?

I'm working on an exercise from functional analysis. Let $E$ be a vector space and $\|\cdot\|_1$ and $\|\cdot\|_2$ be two complete norms on $E$. Now suppose that $E$ satisfies the following property: $\bullet$ if $(x_n)$ is a sequence in $E$ and…
molan
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Hilbert Spaces and Banach Spaces

I have a problem with the definition of Hilbert Space and Banach Space. What is the difference between a Hilbert Space and a Banach Space?
JRR
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Prove that the Lipschitz constant cannot be less than $1$

I was asked to study the following map: let $I=[0,1]$ and let $f(s)=\log(1+s^2)$ for any $s\in\mathbb R$. For every $u\in L^1(I,\mathbb R)$ we set $$(F(u))(x)=\int_0^xf(u(t))\mathrm d t.$$ First of all I had shown that $F$ maps $L^1(I,\mathbb R)$…
uforoboa
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Finding the inner product given the norm

Possible Duplicate: Norms Induced by Inner Products and the Parallelogram Law So suppose we are given a norm on a vector space. If the Parallelogram law holds does that automatically mean we have the inner product which we can find using the…
Adam Rubinson
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Adjoints in Banach spaces

I been trying to figure out if the construction of the isomorphism between $g(W)$ and $f^{*}(W^{*})$ via $$B_V(v, g(w)) = B_W(f(v),w) \quad \forall v \in V, w \in W $$ described (at least this is what I think they mean) in the wikipedia page , is…
user123124
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