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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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Hahn Banach extension of linear functional $f$

Let $f:(c_{00},\|\cdot\|_1)\to \mathbb C $ be a non zero continuous linear functional. The number of Hahn-Banach extensions of f to $(\ell^1,\|\cdot\|_1)$ is one two infinite three I have no idea how to get the result. Hahn Banach theorem on…
user304089
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Continuous with respect to weak convergence implies affine

Let $\phi : \mathbb R \rightarrow \mathbb R$ be a continuous function such that whenever $f_n \rightarrow f$ weakly in $L^2[0,1]$, we have $\phi\circ f_n \rightarrow \phi\circ f$ weakly in $L^2[0,1]$. I am trying to prove that $\phi$ must be an…
user15464
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Subset of sequence space closed

Is the set $E:=\{(x_n)_{n\in \mathbb{N}} \in \ell^{\infty}\ |\; x_i \in \mathbb{C}, \lim_{n \rightarrow \infty } x_n = 0 \}$ closed in $\ell^{\infty}$ equipped with $\lVert (x_n)_{n\in \mathbb{N}} \lVert_{\infty}=\sup_{n\in\mathbb{N}} |x_n|$? I've…
Julian
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Map from a normed space to its double dual.

Let $X$ be a normed space. Let the function $J:X \rightarrow X'' $ be defined by $$J(x)(x')=x'(x)\ \forall\ x'\in X' $$ where $X''=\{f:X'\rightarrow\mathbb{C} \mid \hbox{$f$ is bounded linear} \}$ and $X'$ is the dual of $X$. Is $J$ injective? I…
Epsilon
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Proof that $\lim_{n\to\infty}\|x_n-x\|=0$ with weak convergence

I want to prove that in a normed linear space $H$, that if $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: $$\lim_{n\to\infty}\|x_n-x\|=0$$ Can I please have a hint? Also does $\langle x,x\rangle^{\frac12}=\|x\|$ or…
Functional Analysis
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In the Hahn-Banach theorem, what is the purpose of the 'dominating function'?

I am studying functional analysis by reading "Elements of Functional Analysis" by IJ Maddox (which was the set text for the Open University's now discontinued course on this subject). In the Hahn-Banach theorem, what is the purpose of the…
Jim Clow
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Basis for $l^{\infty}$

As the question stated, we know that $\{e_i\}$ doesn't form a basis for $l^{\infty}$. So how can we find a basis for $l^{\infty}$, no matter it is Schauder or Hamel basis.
89085731
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Assume $T$ is compact operator and $S(I- T) = I $.Is this true that $(I- T)S =I$?

Suppose $S,T \in {\rm B}(X)$ and assume $T$ is compact operator and $S(I- T) = I $.Is this true that $(I- T)S =I$?
Payton-PHP
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Riesz Lemma with $\alpha=1$ and Linear Bounded Functional

Show that on a normed linear space $X$, Riesz lemma with $\alpha=1 $ holds implies that every bounded linear functional attains its norm on the unit sphere of $X$. This is not a homework question and I have tried a huge amount of approaches (too…
user14082
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$\int_a^b |f(x)||g(x)| dx \leq \left(\int_a^b |f(x)|^p dx\right)^{\frac1p}\left(\int_a^b |g(x)|^q dx\right)^{\frac{1}{q}}$

Let $p\gt 1,q\gt 1$ be the dual indices, $\frac1p + \frac1q = 1$ and let $X$ be the space of all continuous functions on $[a,b]$ with two real numbers $a\lt b$. $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ I want to prove that: $$\int_a^b…
Understand
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Is this proof that $c_0$ is a closed subspace of $\ell^\infty$ correct? Also need some help finish it.

$c_0 \subset \ell^\infty$ is the subspace of all sequences of scalars converging to zero. Let $x \in \bar{c_0}$. Then there exists a Cauchy sequence of sequences $(x^k)_{k \in \mathbb{N}}$ in $c_0$, where $x^k = (a_1^k, a_2^k,a_3^k \dots )$ such…
bananaspace
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Let $X$ be a linear normed space, and $L$ a nontrivial linear functional on $X$. Prove the following are equivalent:

Let $X$ be a linear normed space, and $L$ a nontrivial linear functional on $X$. Prove the following are equivalent: $1) L$ is continuous $2)$ The null space of $L$ is proper, closed linear subspace of $X$ $3)$ The null space of $L$ is not dense in…
Extremal
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Prove there exists an element $x\in X$, such that $\|x\|=1$ and $d(x,M)=1$.

If $M$ is a finite dimensional proper subspace of a normed linear space $X$, prove there exists an element $x\in X$, such that $\|x\|=1$ and $d(x,M)=1$. My Try and Problems: If $M$ were closed then I can apply Hahn Banach Theorem. But being finite…
Extremal
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Norm of adjoint

Assume we have 2 injective continuous operators with dense images $A$ and $B$ on a Hilbert space $\mathbb H$ and $B$ is self adjoint. Further let there be constants $a_1$ and $a_2$ such that- $a_1\|Bu\| \leq \|Au\| \leq a_2\|Bu\|$ for all $u \in…
Madhuresh
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discontinuous Linear operator from a Banach space to a normed vector space

How do I find that example of a discontinuous linear operator A from a Banach space to a normed vector space such that A has a closed graph?
Jong hee Kim
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