Hilbert Spaces and Spectral Theory - Functional Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2008

Let \(H = L^2([0, \pi])\) with the usual inner product. For \(n \in \mathbb{N}\), let \[ u_n(t) = \sqrt{\frac{2}{\pi}} \sin nt, \ t \in [0, \pi], \ \text{and} \ E = \{u_n : n \in \mathbb{N}\}. \] Then

A

\(E\) is not a linearly independent subset of \(H\)

B

\(E\) is a linearly independent subset of \(H\), but is not an orthonormal subset of \(H\)

C

\(E\) is an orthonormal subset of \(H\), but is not an orthonormal basis for \(H\)

D

\(E\) is an orthonormal basis for \(H\)

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2

2009 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2009

Let \(\{e_n\}_{n=1}^{\infty}\) be an orthonormal sequence in a Hilbert space \(H\) and let \(x (\neq 0) \in H\). Then

A

\(\lim_{n \to \infty} \langle x, e_n \rangle\) does not exist

B

\(\lim_{n \to \infty} \langle x, e_n \rangle = \|x\|\)

C

\(\lim_{n \to \infty} \langle x, e_n \rangle = 1\)

D

\(\lim_{n \to \infty} \langle x, e_n \rangle = 0\)

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3

2010 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2010

Which one of the following sets of functions is NOT orthogonal (with respect to the \( L^2 \)-inner product) over the given interval?

A

\(\{\sin nx : n \in \mathbb{N}\}, \ -\pi < x < \pi \)

B

\(\{\cos n\pi x : n \in \mathbb{N}\}, \ -\pi < x < \pi \)

C

\(\{x^{2n-\frac{1}{2}} : n \in \mathbb{N}\}, \ -1 < x < 1 \)

D

\(\{x^{2n+1} : n \in \mathbb{N}\}, \ -1 < x < 1 \)

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4

2010 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2010

Which one of the following statements is correct?

A

Both \(X_0 \) and \(X_0^\perp \) are complete

B

Neither \(X_0 \) nor \(X_0^\perp \) is complete

C

\(X_0 \) is complete but \(X_0^\perp \) is not complete

D

\(X_0^\perp \) is complete but \(X_0 \) is not complete

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5

2010 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2010

Let \( y(t) = t^3 \), \( t \in [0,1] \) and \( x_0 \in X_0^\perp \) be the best approximation of \( y \). Then \( x_0(t) \), \( t \in [0,1] \), is

A

\(\frac{4}{5} t^2 \)

B

\(\frac{5}{6} t^2 \)

C

\(\frac{6}{7} t^2 \)

D

\(\frac{7}{8} t^2 \)

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6

2011 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2011

Let \( H \) be a complex Hilbert space and \( H^* \) be its dual. The mapping \( \phi : H \to H^* \) defined by \( \phi(y) = f_y \) where \( f_y(x) = \langle x, y \rangle \) is

A

not linear but onto

B

both linear and onto

C

linear but not onto

D

neither linear nor onto

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7

2011 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2011

Let \( e_i = (0, \ldots, 0, 1, 0, \ldots) \) (i.e., \( e_i \) is the vector with 1 at the \( i^{th} \) place and 0 elsewhere) for \( i = 1, 2, \ldots \). Consider the statements:
P: \( \{ f(e_i) \} \) converges for every continuous linear functional on \( l^2 \).
Q: \( \{ e_i \} \) converges in \( l^2 \).
Then, which of the following holds?

A

Both P and Q are TRUE

B

P is TRUE but Q is not TRUE

C

P is not TRUE but Q is TRUE

D

Neither P nor Q is TRUE

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8

2012 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2012

Let \(H\) be a Hilbert space and \(S^{\perp}\) denote the orthogonal complement of a set \(S \subseteq H\). Which of the following is INCORRECT?

A

For \(S_1, S_2 \subseteq H ; S_1 \subseteq S_2 \Rightarrow S_1^{\perp} \subseteq S_2^{\perp}\)

B

\(S \subseteq (S^{\perp})^{\perp}\)

C

\(\{0\}^{\perp} = H\)

D

\(S^{\perp}\) is always closed.

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9

2012 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2012

Let \(H\) be a complex Hilbert space, \(T : H \to H\) be a bounded linear operator and let \(T^*\) denote the adjoint of \(T\). Which of the following statements are always TRUE?
P: \(\forall x, y \in H, \langle Tx, y \rangle = \langle x, T^* y \rangle\)
Q: \(\forall x, y \in H, \langle x, Ty \rangle = \langle T^* x, y \rangle\)
R: \(\forall x, y \in H, \langle x, Ty \rangle = \langle x, T^* y \rangle\)
S: \(\forall x, y \in H, \langle Tx, Ty \rangle = \langle T^* x, T^* y \rangle\)

A

P and Q

B

P and R

C

Q and S

D

P and S

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10

2013 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2013

Let \(\mathcal{H}\) be a Hilbert space and let \(\{e_n : n \ge 1\}\) be an orthonormal basis of \(\mathcal{H}\). Suppose \(T: \mathcal{H} \to \mathcal{H}\) is a bounded linear operator. Which of the following CANNOT be true?

A

\(T(e_n) = e_1\) for all \(n \ge 1\)

B

\(T(e_n) = e_{n+1}\) for all \(n \ge 1\)

C

\(T(e_n) = \sqrt{\frac{n+1}{n}} e_n\) for all \(n \ge 1\)

D

\(T(e_n) = e_{n-1}\) for all \(n \ge 2\) and \(T(e_1) = 0\)

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11

2015 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2015

Let \(V\) be a closed subspace of \(L^2[0, 1]\) and let \(f, g \in L^2[0, 1]\) be given by \(f(x) = x\) and \(g(x) = x^2\). If \(V^\perp = \text{Span}\{f\}\) and \(Pg\) is the orthogonal projection of \(g\) on \(V\), then \((g - Pg)(x), x \in [0, 1]\), is

A

\(\frac{3}{4}x\)

B

\(\frac{1}{4}x\)

C

\(\frac{3}{4}x^2\)

D

\(\frac{1}{4}x^2\)

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12

2017 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2017

Let \(\{e_n:n\in\mathbb{N}\}\) be an orthonormal basis of a Hilbert space \(H\). Let \(T:H\to H\) be given by \(Tx=\sum_{n=1}^{\infty}\frac{1}{n}\langle x,e_n\rangle e_n\). For each \(n\in\mathbb{N}\), define \(T_n:H\to H\) by \(T_nx=\sum_{j=1}^{n}\frac{1}{j}\langle x,e_j\rangle e_j\). Then

A

\(\|T_n-T\|\to 0\) as \(n\to\infty\)

B

\(\|T_n-T\|\not\to 0\) as \(n\to\infty\) but for each \(x\in H\), \(\|T_nx-Tx\|\to 0\) as \(n\to\infty\)

C

for each \(x\in H\), \(\|T_nx-Tx\|\to 0\) as \(n\to\infty\) but the sequence \((\|T_n\|)\) is unbounded

D

there exist \(x, y\in H\) such that \(\langle T_nx, y\rangle\not\to\langle Tx, y\rangle\) as \(n\to\infty\)

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13

2018 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2018

Let \(L^2([0,1])\) be the Hilbert space of all real valued square integrable functions on \([0,1]\) with the usual inner product. Let ϕ be the linear functional on \(L^2([0,1])\) defined by \[ϕ(f) = \int_{1/4}^{3/4} 3\sqrt{2} f \, dμ,\] where μ denotes the Lebesgue measure on \([0,1]\). Then \(\| ϕ \| = ________.\)

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14

2018 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2018

Let \(U\) be an orthonormal set in a Hilbert space \(H\) and let \(x \in H\) be such that \(\|x\| = 2\). Consider the set \[E = \left\{ u \in U : |\langle x, u \rangle| \geq \frac{1}{4} \right\}.\] Then the maximum possible number of elements in \(E\) is ________.

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15

2020 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2020

Let \(\{e_n\}_{n=1}^{\infty}\) be an orthonormal basis for a separable Hilbert space \(H\) with the inner product \(\langle \cdot, \cdot \rangle\). Define
\[f_n = e_n - \frac{1}{n+1} e_{n+1} \text{ for } n \in \mathbb{N}.\]
Then
(A) the closure of the span \(\{f_n : n \in \mathbb{N}\}\) equals \(H\)
(B) \(f = 0\) if \(\langle f, f_n \rangle = \langle f, e_n \rangle\) for all \(n \in \mathbb{N}\)
(C) \(\{f_n\}_{n=1}^{\infty}\) is an orthogonal subset of \(H\)
(D) there does not exist nonzero \(f \in H\) such that \(\langle f, e_2 \rangle = \langle f, f_2 \rangle\)

A

the closure of the span \(\{f_n : n \in \mathbb{N}\}\) equals \(H\)

B

\(f = 0\) if \(\langle f, f_n \rangle = \langle f, e_n \rangle\) for all \(n \in \mathbb{N}\)

C

\(\{f_n\}_{n=1}^{\infty}\) is an orthogonal subset of \(H\)

D

there does not exist nonzero \(f \in H\) such that \(\langle f, e_2 \rangle = \langle f, f_2 \rangle\)

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16

2021 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2021

Let \(\{e_n : n = 1,2,3,...\}\) be an orthonormal basis of a complex Hilbert space \(H\). Consider the following statements:

P: There exists a bounded linear functional \(f: H \to \mathbb{C}\) such that \(f(e_n) = \frac{1}{n}\) for \(n = 1,2,3,...\).

Q: There exists a bounded linear functional \(g: H \to \mathbb{C}\) such that \(g(e_n) = \frac{1}{\sqrt{n}}\) for \(n = 1,2,3,...\).

Then

A

both P and Q are TRUE

B

P is TRUE and Q is FALSE

C

P is FALSE and Q is TRUE

D

both P and Q are FALSE

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17

2021 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2021

Let \(H\) be a complex Hilbert space. Let \(u, v \in H\) be such that \((u, v) = 2\). Then \(\frac{1}{2\pi} \int_0^{2\pi} \|u + e^{it}v\|^2 e^{it} dt = ________\).

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18

2021 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2021

Let $L^2[-1, 1]$ be the Hilbert space of real valued square integrable functions on $[-1, 1]$ equipped with the norm $\|f\| = \left( \int_{-1}^{1} |f(x)|^2 dx \right)^{1/2}$.
Consider the subspace $M = \{ f \in L^2[-1, 1] : \int_{-1}^{1} f(x) dx = 0 \}$.
For $f(x) = x^2$, define $d = \inf \{ \|f - g\| : g \in M \}$. Then

A

$d = \frac{\sqrt{2}}{3}$

B

$d = \frac{2}{3}$

C

$d = \frac{3}{\sqrt{2}}$

D

$d = \frac{3}{2}$

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19

2022 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2022

Let \(\{e_k : k \in \mathbb{N}\}\) be an orthonormal basis for a Hilbert space \(H\). Define \(f_k = e_k + e_{k+1}, k \in \mathbb{N}\) and \(g_j = \sum_{n=1}^{j} (-1)^{n+1} e_n, j \in \mathbb{N}\). Then \(\sum_{k=1}^{\infty} |\langle g_j, f_k \rangle|^2 =\)

A

0

B

\(j^2\)

C

\(4j^2\)

D

1

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20

2022 · Mathematics · Functional Analysis · Hilbert Spaces and Spectral Theory

Mathematics (MA) 2022

Let \(T : L^2[-1,1] \to L^2[-1,1]\) be defined by \(Tf = \hat{f}\), where \(\hat{f}(x) = f(-x)\) almost everywhere. If \(M\) is the kernel of \(I - T\), then the distance between the function \(\phi(t) = e^t\) and \(M\) is

A

\(\frac{1}{2}\sqrt{(e^2 - e^{-2} + 4)}\)

B

\(\frac{1}{2}\sqrt{(e^2 - e^{-2} - 2)}\)

C

\(\frac{1}{2}\sqrt{(e^2 - 4)}\)

D

\(\frac{1}{2}\sqrt{(e^2 - e^{-2} - 4)}\)

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