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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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Find a norm of operator form l1 to l1

I have an operator $A: \ell_1 \to \ell_1, Ax = (x_1+x_2, x_1-x_2, x_3,...,x_k,...)$ AFAIK, norm of $\ell_1$ is $\sum_{n=1}^{\infty}|x_n|$ How to find a norm of this operator?
michaeluskov
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Does this contradict the Closed Graph Theorem?

Question: Let $C^{'}[0,1]$ and $C[0,1]$ be endowed with sup norm. Define $T:C^{'}[0,1]\to C[0,1]$ by $$Tf=f^{'}\text{ for each }f\in C^{'}[0,1]$$ Where, $'$, indicates differentiation. Prove that $T$ is a linear map with closed graph but it is not…
Hassan Muhammad
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This map $f$ is not continuous with respect to sup norm

Question: On the space $\ell^1$ for $x=(\alpha_1,\alpha_2,\ldots)\in{\ell^1}$, define $$f(x)=\sum\limits_{n=1}^\infty \alpha_n$$ Prove that $f$ is not continuous with respect to $\|x\|_\infty =\sup_n|\alpha_n|$. This is my proof: Since $f$ is a…
Hassan Muhammad
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Let $X$ be a reflexive Banach space. $T\in\mathcal{L}(X,X)\,\Longleftrightarrow\, $if $x_n\rightharpoonup x$, then $T(x_n)\rightharpoonup T(x)$

I have the forward direction: $(\Longrightarrow)$ Let $T\in\mathcal{L}(X,X)$ and let $f\in X^*$. Since both $T$ and $f$ are bounded, then both $T$ and $f$ are continuous in the norm topology. Then $f\circ T$ is continuous with respect to the norm…
Laars Helenius
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Does homeomorphism imply an equivalent norm?

Can we find a banach space X on which there are two non-equivalent norms, but they induce the same topology? I am almost sure that there should be such example, otherwise it would seem to be a rather strong and surprising result. (Note that we…
user113991
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Equicontinuous set

Let $\mathcal E$ be the set of all functions $u\in C^1([0,2])$ such that $u(x)\geq 0$ for every $x\in[0,2]$ and $|u'(x)+u^2(x)|<1$ for every $x\in [0,2]$. Prove that the set $\mathcal F:=\{u_{|[1,2]}: u\in\mathcal E\}$ is an equicontinuous subset of…
uforoboa
  • 7,074
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Norm of diagonal operator

Let $\{e_n\}$ be the usual basis for $l^2$ and $\{\alpha_n\}$ be a bounded sequence of scalars. For all n, define $Ae_n=\alpha_n e_n$ on $l^2$. Show that $||A||=\sup\alpha_n$. I can show $||A||\leq\sup\alpha_n$ easily. My problem is showing…
nika
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Do Incomplete Normed Vector Spaces Whose Duals Are Reflexive Exist?

It is clear to me that if $X$ is a Banach space and its dual $X^*$ is reflexive, then $X$ is also reflexive (that is, the natural map between $X$ and its double dual $X^{**}$ is a surjective isometric isomorphism). However, I suspect that the…
triple_sec
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Calculating square roots of operators using power series for $\sqrt{1 - z}$

The following is a theorem from Reed & Simon's Methods of Modern Mathematical Physics, Volume I. Here, we are working in a complex Hilbert space $(\mathcal{H}, (\cdot, \cdot))$, and $\mathscr{L}(\mathcal{H})$ is the space of bounded linear operators…
JZS
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A use of Hahn-Banach and Riesz Representation

Let $X$ be a compact Hausdorff topological space. Suppose $X$ is not a singleton set and $C(X)$ denotes the space of continuous functions on $X$. Do we have that for all $L \subset C(X)$ a nondense subspace, there exist two probability measures…
Jeff
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An equicontinuity condition connected with the derivative matrix

In the book "Markov Chains and Stochastic Stability" (page 171, http://probability.ca/MT/) of Meyn and Tweedie there is used the following condition of equicontinuity: Assume that functions $f_n:\mathbb{R}^d\rightarrow\mathbb {R}$, $n\in\mathbb{N}$…
Dawid C.
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If $\sum_n \|x\|< \infty$, how to show that $\sum x_n$ is convergent in the Hilbert space $H$.

Let $\{x_n\}$ be a sequence in a Hilbert space $H$. If $\sum_n \|x\|< \infty$, how to show that $\sum x_n$ is convergent in $H$? There is no doubt that $x_n \rightarrow 0$ as $n \rightarrow \infty$ (right?) since we have that \begin{align*} \sum_n…
Ingvar
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$X$ be a reflexive space then show that $X$ is Banach Space and is reflexive in any equivalent norm.

Let $X$ be a reflexive space then show that $X$ is Banach Space and is reflexive in any equivalent norm. ...................................................................... I am trying it in a following way... Since dual $X^{'}$ of any normed…
mathonlymath
  • 177
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Proving that the graph of a operator is closed

Let $E$ be a Banach space and let $T:E\mapsto E'$ be a linear operato satisfying $\langle Tx,x\rangle\geq0$ for all $x\in E$. How to prove that the graph of $T$ is closed?
Thom Kostova
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Solving equations where the solution is an operator

Ok, so here's some context. Solving regular equations we might have something like this: $2 + x = 5$, solving for $x$ we get 3. We might even have an equation like $x + y = 5$ where there are multiple solutions. But what's in common with all these…
Luka Horvat
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