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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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A property of $l_p$ space

I'd like to show the following. Let $(u_n)_{n\geq 1}$ be a sequence of elements in $l_p(\mathbb{N})$. Suppose $u_n$ is p-norm-bounded by some $m>0$ and that for every $k\geq 1$, $\lim\limits_{n\to\infty}u_n(k)=0$, then $u_n$ has limit $0$ in weak…
Rodriguez
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An application of Hahn-Banach: There exists non-zero linear forms vanishing on a subspace

I have to prove the following: Let $E$ be a normed vector space, and $F$ a proper subspace of $E$. Prove that there exists a continuous, non-zero linear form $L$ on $E$ which vanishes on $F$. This seemed easy at first: I'll define $\ell$ to be…
Jack M
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Conway's proof of Goldstine's theorem.

In proving Goldstine's theorem, Conway states the following. Suppose $B$ is the weak-star closure of $J(B_X)$ in $B_{X''}$, and assume there is some $x_0'' \in B_{X''}\smallsetminus B$. He claims Hahn-Banach implies there is some $x'\in X'$ and…
Pedro
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Pointwise convergence if and only if $\|f_n-f\| \to 0$

Is there a norm in the C[0,1] space function such that this happens? $ f_n(x) \rightarrow f(x) $ if and only if $ \|f_n - f\| \rightarrow 0$ Whene $f_n$ is a function sucession, and $ f_n(x) \rightarrow f(x) $ means Pointwise convergence
Maria
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Hellinger-Toeplitz Theorem and Uniform Boundedness Principle

The question is prove the Hellinger-Toeplitz theorem, using the uniform boundedness principle. All hints are welcome
JRR
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Universal Property of Basis

Let $X$ be a $\mathbb{K}$-linear space. $E\subseteq X$. Then $E$ is a basis of $X$ $\iff$ for every $\mathbb{K}$-linear space $Y$ and for every $f:E\rightarrow Y$, there exists a unique $\mathbb{K}$-linear extension $T:X\rightarrow Y$ of $f$. Is…
mmcrjx
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Convergence in quotient space $X/Y$

Let $Y$ is closed subspace of $X$ and $(x_n+Y)\rightarrow (x+Y)$ in $X/Y$ for some $x_n,x\in X$. Question is to see that there exist $y_n\in Y$ such that $x_n+y_n\rightarrow x$. This seems to be true but i could not give a proof. Given $\epsilon>0$…
user312648
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A question about Banach spaces

Can we find a norm that makes the space of all real sequences a Banach space?
Marios Gretsas
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Bounded integral operator on $L^p$

This is an exercise on Stein's Real Analysis. Suppose $w$ is a measurable function with $0 < w < \infty$ almost everywhere, and $K$ a measurable function on $\mathbb R^{2d}$ that satisfies (i) $\displaystyle\int |K(x,y)|w(y) dy \leq Aw(x)$ for…
P. Factor
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Question on Rudin's functional analysis Theorem 4.9

$M$ is a closed subspace of a Banach space $X$ part (a) of theorem 4.9 proves $M^*=X^*/M^{\perp}$ ("=" means isometric isomorphism) He constructed a map $\sigma:M^*\to X^*/M^{\perp}$ $$\sigma (m^*)=x^*+M^{\perp}$$ and has already proved $\sigma$ is…
Lookout
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Is $C^{k+1}(X)$ compactly contained in $C^{k}(X)$ for a closed manifold $X$?

Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let $C^k(X)$ denote the space of $k$-times continuously differentiable functions equipped with the C$^k$ norm. Is $C^{k+1}(X)$ compactly contained in $C^k(X)$? Does this follow from…
trex
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Derivative operator on polynomial space $P[0,1]$.

Let $P[0,1]$ be the space of real polynomial function on$[0,1]$ with $\|p\|= sup\{|p(x)|:0\leq x \leq 1\}. T(p)(x)=\frac{d}{dx}(p(x)).$ Then it is clear that $T$ is not a bounded operator. How to discuss the closeness of this operator. Is it close…
user300712
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Dual of forward difference operator

For simplicity, consider the bounded linear operator $L:\ell^\infty\rightarrow \ell^\infty$ defined by $$(Lx)_k=x_{k+1}-x_k,$$ where here, $\ell^\infty$ consists of all bounded bi-infinite sequences of real numbers (ie. indexed by $\mathbb{Z}$). In…
Kev C
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An operator has closed range iff some condition holds

If $T:X \rightarrow Y$ is a bounded linear operator that $T^{-1}(0)=\{0\}$ then $\mathcal{R}(T)=\{Tx\}_{x \in X}$ is closed $\Leftrightarrow$ there is no sequence $(x_n)$ with $||x_n||=1$ such that $Tx_n \rightarrow 0$ I have $\Rightarrow$ but no…
xyz
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Find the bounds for the norm $T:l_2 \to l_2$.

Given $T:l_2 \to l_2$ define as $T((x_1,x_2,\ldots,x_n,\ldots))=(x_2-x_1,x_3-x_2,\ldots,x_{n+1}-x_n,\ldots)$ then which of the following is true, $\|T\|=1$ $\|T\|\geq2$ $1<\|T\|\leq2$ None of above. What I did- I used $\|T\|^2=\langle…
Harry Potter
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