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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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Absolute series convergence of linear functional values implies a kind of sequence value series convergence in Banach spaces.

Ran across this question, and am not sure how to tackle it. Let $X$ be a Banach space, and let $\{x_n\}$ be a sequence in $X$ such that $\sum_{n=1}^{\infty} |\phi(x_n)|$ converges for all $\phi \in X^*$, the dual of $X$. Show that…
Johnny Apple
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Rigourously proving continuity of multiplication of vector with scalar

Let $X$ be vector space and $q : X \mapsto \mathbb {R}$ function with the following properties: $q(x) \geq 0$ $\forall x \in X$ and $q(x)=0 \Leftrightarrow x=0$ $q(x+y) \leq q(x)+q(y)$ $\forall x,y \in X$ $q(\alpha x)\leq q(x)$ $\forall x \in X$…
ters
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If a subspace contains a dense subset of a unit ball, then the subspace also contains a dense subset of the whole space

Suppose $X$ is a normed space. Denote $B_X$ as closed unit ball in $X$. Let $D$ be a dense subset of $B_X$ and $Y$ contains $D$ where $Y$ is a subspace of $X$. Show that $Y$ contains a dense subset $E$ of $X$. My attempt: Suppose that $E$ is a dense…
Idonknow
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Is $F=\{g\in\mathcal C^1([0,1])\mid \exists x\in [0,1]: (f(x),f'(x))=(0,0)\}$ open or close in $\mathcal C^1([0,1])$?

Consider $(\mathcal C^1([0,1]),\|\cdot \|)$ where $$\|f \|=\|f\|_\infty +\|f'\|_\infty $$ where $\|g\|_\infty :=\sup_{[0,1]}|g|.$ Let $$F=\{g\in \mathcal C^1([0,1])\mid \exists x\in [0,1], (f(x),f'(x))=(0,0)\}.$$ 1) Is $F$ open or close in…
idm
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Show that a weak-to-norm continuous operator is compact

The following screenshot is taken from the book 'Topics in Banach Space Theory'. I don't understand how the author obtains the second sentence, which states that 'Therefore in order to prove that $T$ is compact it suffices to show that $T|_{B_X}$…
Idonknow
  • 15,643
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Show that in $L_1(\mathbb{R})$ every absolutely convergent series converges.

Show that in $L_1(\mathbb{R})$ every absolutely convergent series converges. My try: Take some absolutely convergent series in $L_1(\mathbb{R})$. Then we have: $$ \sum_{n=0}^\infty \Vert f_n\Vert = \sum_{n=0}^\infty \int_{\mathbb{R}} |f_n|\…
user161518
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Does a linear operator preserve compact support?

I have a question in my mind. I would like to know whether a linear operator actually preserves compact support? Or is there any relation? Does it make sense to ask this question, at all?
Theorem
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The norm-closed unit ball of $c_0$ is not weakly compact

Show that the norm-closed unit ball of $c_0$ is not weakly compact; recall that $c_0^*=\ell_1$.
Ali Qurbani
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How do I prove that $I+T^*T$ is invertible?

Let $T$ be a bounded linear operator in a Hilbert space, $T^*$ the adjoint of $T$. Then how to show that $I + T^*T$ is invertible? Thanks.
Freeman
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How is Hausdorff property of $X$ used in the theorem regarding completeness of $C(X)$?

I'm trying to understand a proof of the following theorem (from Banach Algebra Techniques in Operator Theory by Douglas): Here is my question: Where is the Hausdorff property used in the proof?
user9464
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Show that functional $f(x)=x'((a+b)/2)$ is bounded in $C^1([a,b])$ but unbounded in $C([a,b])$

Let $C^1([a,b])$ be normed vector space of all continuously differentiable functions defined on $[a,b]$ with a norm $$||x||=\sup_{t \in [a,b]} |x(t)|+\sup_{t \in [a,b]} |x'(t)|$$ We have to show that the functional $f(x)=x'((a+b)/2)$ is bounded on…
Boby
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Linear map $L : C([0,1])\rightarrow C([0,1])$ continuous?

Let $X=C([0,1])$ equipped with the norm $\Vert\cdot \Vert=\max_{x\in [0,1]}|f(x)|$. If $L:X\rightarrow X$ is linear, is $L$ continuous? If not, what if $Lf\ge 0$ for $f\ge0$ for $\forall x \in [0,1]$ is assumed? Edit: Rephrased the…
Stanley5664
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Existence of discontinuous linear functional on arbitrary infinite dimensional normed vector space without Axiom of Choice

The title says it all. Given an arbitrary infinite dimensional normed vector space $X$, can you show there exists a discontinuous (perhaps unbounded) linear functional on $X$ without resorting to Axiom of Choice/Hamel basis techniques? I know…
Toeplitz
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How do I prove the norm of this linear functional is $2$?

$f$ is defined on $C[-1, 1]$ by $$f(x)=\int_{-1}^0 x(t) dt - \int_0^1 x(t) dt.$$ I can show that $\|f\| \le 2$. I don't know how to show $\|f\| \ge 2$.
charlie
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Is $X$ separable iff $S_X=\{x\in X \mid ||x||=1\}$ separable?

While going through this question in math stackexchange I have come across the fact that "if $(X,d)$ is a separable metric space, then for any subset $Y\subseteq X$, the subspace $(Y,d)$ is separable" which is mentioned in the answer by 1015 to the…
usermath
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