Semi-inner product

The semi-inner product is a term from the mathematical branch of functional analysis . It is defined for - vector spaces , where stands for the field of real or complex numbers, and generalizes the concept of the inner product . ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} }$

definition

A semi-inner product on a vector space is a map with the following properties ${\displaystyle \mathbb {K} }$ ${\displaystyle V}$ ${\displaystyle [\cdot ,\cdot ]:\,V\times V\rightarrow \mathbb {K} }$

1. ${\displaystyle x\mapsto [x,y]}$is a linear functional for each . ${\displaystyle y\in V}$
2. ${\displaystyle [x,x]\,>\,0}$for all of 0 different . ${\displaystyle x\in V}$
3. ${\displaystyle |[x,y]|^{2}\,\leq \,[x,x][y,y]}$for everyone . ${\displaystyle x,y\in V}$

Comparison with internal products

If there is an inner product on the vector space , this trivially fulfills the first two of the above conditions, and the Cauchy-Schwarz inequality shows that the third is also fulfilled. Therefore every inner product is a semi-inner product. ${\displaystyle \langle \cdot ,\cdot \rangle :V\times V\rightarrow \mathbb {K} }$ ${\displaystyle V}$

The reverse is not true. What the semi-inner product lacks in order to be an inner product is the hermiticity and the linearity or sesquilinearity in the second argument.

Standardized spaces

If a semi-inner product is on a vector space , this becomes a normalized space by definition . Conversely, one can show that every standardized space is created in this way by a semi-internal product, that is, for every norm there is a semi-internal product, so that the above relationship applies. That was the motivation for G. Lumer to introduce this term. This has by far not the same meaning as the inner product , but allows in some situations to transfer Hilbert space arguments to Banach spaces. ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle V}$ ${\displaystyle \|x\|:=[x,x]^{1/2}}$

A semi-inner product of a normalized space, that is, one that represents the given norm by the above formula, is generally ambiguous. One can show that one can always choose one that is homogeneous in the second argument, that is, for which applies to all and . The dash stands for the complex conjugation, which does not apply in the case of real vector spaces. ${\displaystyle [x,\lambda y]={\overline {\lambda }}[x,y]}$ ${\displaystyle x,y\in V}$ ${\displaystyle \lambda \in \mathbb {K} }$

Examples

• The norm on Hilbert spaces is given by a scalar product ; the norm on an inner product space is more general than the norm induced by the inner product .
• L ^p -spaces : If is a measure space and is , then set for : ${\displaystyle (X,\mu )}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle x,y\in L^{p}(X,\mu )}$

${\displaystyle [x,y]:={\frac {1}{\|y\|_{p}^{p-2}}}\int _{X}x|y|^{p-1}\mathrm {sgn} \,y\,\mathrm {d} \mu }$

This is a semi-internal product that defines the standard on . ${\displaystyle p}$ ${\displaystyle L^{p}(X,\mu )}$

Continuity properties

Let it be the set of all vectors of norm 1 of a normalized vector space. A semi-inner product to a standardized space is called continuous , if for all , Re stands for the formation of the real part . Caution is advised with this term, because it does not mean that the semi-inner product is continuous as a mapping , the above continuity property is obviously much weaker. One says that the semi-inner product is uniformly continuous if the above Limes equation persists uniformly on the set . ${\displaystyle S}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle \lim _{\lambda \to 0}\mathrm {Re} ([y,x+\lambda y])=\mathrm {Re} ([y,x])}$ ${\displaystyle x,y\in S}$ ${\displaystyle V\times V\rightarrow \mathbb {K} }$ ${\displaystyle \{(x,y);x,y\in S\}}$

These continuity properties can be related to the differentiability properties of the standard. A standardized space is called Gâteaux -differentiable , if

${\displaystyle \lim _{\lambda \to 0}{\frac {\|x+\lambda y\|-\|x\|}{\lambda }}}$

exists for all , and equally Fréchet -differentiable , if this limit exists equally on . ${\displaystyle x,y\in S}$ ${\displaystyle \{(x,y)\mid x,y\in S\}}$

The following sentence applies:

• A semi-inner product is continuous (or uniformly continuous) if and only if the norm is Gâteaux-differentiable (or uniformly Fréchet-differentiable).

The dual space

For a certain class of Banach spaces, a theorem analogous to the representation theorem of Fréchet-Riesz can be proven:

• If there is a uniformly convex Banach space with a continuous semi-inner product , then for every continuous, linear functional there is exactly one with for all . ${\displaystyle V}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle f}$ ${\displaystyle V}$ ${\displaystyle y\in V}$ ${\displaystyle f(x)=[x,y]}$ ${\displaystyle x\in V}$

Of course, one cannot conclude from this, as in the case of the Hilbert spaces, that it is isomorphic to its dual space , because the assignment in the above theorem is generally not linear. In the above example of the -spaces there is a uniformly convex Banach space with a continuous semi-inner product. Every continuous linear functional has the form with a . The one appearing in the above integral is an element from , where . This is then nothing other than the usual duality of spaces . ${\displaystyle V}$ ${\displaystyle f\mapsto y}$ ${\displaystyle L^{p}}$ ${\displaystyle x\mapsto [x,y]}$ ${\displaystyle y\in L^{p}(X,\mu )}$ ${\displaystyle |y|^{p-1}}$ ${\displaystyle L^{q}(X,\mu )}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle L^{p}}$

Numerical range of values

The numerical range of values ​​of a linear operator in a normalized space can be described using an associated semi-inner product . The numerical range of is the end of the convex hull of the set . ${\displaystyle T}$ ${\displaystyle V}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle T}$ ${\displaystyle \{[Tx,x];x\in V,[x,x]=1\}}$

Individual evidence

1. ^ G. Lumer: Semi-inner Product Spaces , Transactions of the American Mathematical Society, Volume 100, number. 1. (1961), pp. 29-43
2. ^ G. Lumer: Semi-inner Product Spaces , Transactions of the American Mathematical Society, Volume 100, number. 1. (1961), pp. 29-43, Theorem 2
3. ↑ JR Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129: 436-446 (1967), Theorem 1
4. ↑ JR Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129: 436-446 (1967), Theorem 3
5. ↑ JR Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129: 436-446 (1967), Theorem 6
6. ^ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Cambridge University Press (1971), ISBN 0-521-07988-8